6 polynomial expressions and operations

Polynomial Expressions

A mathematics expression is a calculation procedure written in numbers, variables, and operation symbols.

Polynomial Expressions

Example A.

2 + 3x

A mathematics expression is a calculation procedure written in numbers, variables, and operation symbols.

Polynomial Expressions

Example A.

2 + 3x “the sum of 2 and 3 times x”

A mathematics expression is a calculation procedure written in numbers, variables, and operation symbols.

Polynomial Expressions

Example A.

2 + 3x “the sum of 2 and 3 times x” 4x2 – 5x

A mathematics expression is a calculation procedure written in numbers, variables, and operation symbols.

Polynomial Expressions

Example A.

2 + 3x “the sum of 2 and 3 times x” 4x2 – 5x “the difference between 4 times the square of x and 5 times x”

A mathematics expression is a calculation procedure written in numbers, variables, and operation symbols.

Polynomial Expressions

Example A.

2 + 3x “the sum of 2 and 3 times x” 4x2 – 5x “the difference between 4 times the square of x and 5 times x” (3 – 2x)2

A mathematics expression is a calculation procedure written in numbers, variables, and operation symbols.

Polynomial Expressions

Example A.

2 + 3x “the sum of 2 and 3 times x” 4x2 – 5x “the difference between 4 times the square of x and 5 times x” (3 – 2x)2 “the square of the difference of 3 and twice x”

A mathematics expression is a calculation procedure written in numbers, variables, and operation symbols.

Polynomial Expressions

Example A.

2 + 3x “the sum of 2 and 3 times x” 4x2 – 5x “the difference between 4 times the square of x and 5 times x” (3 – 2x)2 “the square of the difference of 3 and twice x”

A mathematics expression is a calculation procedure written in numbers, variables, and operation symbols.

An expression of the form #xN, where the exponent N is a non-negative integer and # is a number, is called a monomial (one-term).

Polynomial Expressions

Example A.

2 + 3x “the sum of 2 and 3 times x” 4x2 – 5x “the difference between 4 times the square of x and 5 times x” (3 – 2x)2 “the square of the difference of 3 and twice x”

A mathematics expression is a calculation procedure written in numbers, variables, and operation symbols.

An expression of the form #xN, where the exponent N is a non-negative integer and # is a number, is called a monomial (one-term). For example, 3x2, –4x3, and 5x6 are monomials.

Polynomial Expressions

Example A.

2 + 3x “the sum of 2 and 3 times x” 4x2 – 5x “the difference between 4 times the square of x and 5 times x” (3 – 2x)2 “the square of the difference of 3 and twice x”

A mathematics expression is a calculation procedure written in numbers, variables, and operation symbols.

Example B. Evaluate the monomials if y = –4 a. 3y2

An expression of the form #xN, where the exponent N is a non-negative integer and # is a number, is called a monomial (one-term). For example, 3x2, –4x3, and 5x6 are monomials.

Polynomial Expressions

Example A.

2 + 3x “the sum of 2 and 3 times x” 4x2 – 5x “the difference between 4 times the square of x and 5 times x” (3 – 2x)2 “the square of the difference of 3 and twice x”

A mathematics expression is a calculation procedure written in numbers, variables, and operation symbols.

Example B. Evaluate the monomials if y = –4 a. 3y2 3y2 3(–4)2

An expression of the form #xN, where the exponent N is a non-negative integer and # is a number, is called a monomial (one-term). For example, 3x2, –4x3, and 5x6 are monomials.

Polynomial Expressions

Example A.

2 + 3x “the sum of 2 and 3 times x” 4x2 – 5x “the difference between 4 times the square of x and 5 times x” (3 – 2x)2 “the square of the difference of 3 and twice x”

A mathematics expression is a calculation procedure written in numbers, variables, and operation symbols.

Example B. Evaluate the monomials if y = –4 a. 3y2 3y2 3(–4)2 = 3(16) = 48

An expression of the form #xN, where the exponent N is a non-negative integer and # is a number, is called a monomial (one-term). For example, 3x2, –4x3, and 5x6 are monomials.

Polynomial Expressions

b. –3y2 (y = –4)

Polynomial Expressions

b. –3y2 (y = –4) –3y2 –3(–4)2

Polynomial Expressions

b. –3y2 (y = –4) –3y2 –3(–4)2 = –3(16) = –48.

Polynomial Expressions

b. –3y2 (y = –4) –3y2 –3(–4)2 = –3(16) = –48.

c. –3y3

Polynomial Expressions

b. –3y2 (y = –4) –3y2 –3(–4)2 = –3(16) = –48.

c. –3y3

–3y3 – 3(–4)3

Polynomial Expressions

b. –3y2 (y = –4) –3y2 –3(–4)2 = –3(16) = –48.

c. –3y3

–3y3 – 3(–4)3 = – 3(–64)

Polynomial Expressions

b. –3y2 (y = –4) –3y2 –3(–4)2 = –3(16) = –48.

c. –3y3

–3y3 – 3(–4)3 = – 3(–64) = 192

Polynomial Expressions

b. –3y2 (y = –4) –3y2 –3(–4)2 = –3(16) = –48.

c. –3y3

–3y3 – 3(–4)3 = – 3(–64) = 192

Polynomial Expressions

Polynomial Expressions

b. –3y2 (y = –4) –3y2 –3(–4)2 = –3(16) = –48.

c. –3y3

–3y3 – 3(–4)3 = – 3(–64) = 192

The sum of monomials are called polynomials (many-terms), these are expressions of the form #xN ± #xN-1 ± … ± #x1 ± #where # can be any number.

Polynomial Expressions

Polynomial Expressions

b. –3y2 (y = –4) –3y2 –3(–4)2 = –3(16) = –48.

c. –3y3

–3y3 – 3(–4)3 = – 3(–64) = 192

The sum of monomials are called polynomials (many-terms), these are expressions of the form #xN ± #xN-1 ± … ± #x1 ± #where # can be any number.

For example, 4x + 7,

Polynomial Expressions

Polynomial Expressions

b. –3y2 (y = –4) –3y2 –3(–4)2 = –3(16) = –48.

c. –3y3

–3y3 – 3(–4)3 = – 3(–64) = 192

The sum of monomials are called polynomials (many-terms), these are expressions of the form #xN ± #xN-1 ± … ± #x1 ± #where # can be any number.

For example, 4x + 7, –3x2 – 4x + 7,

Polynomial Expressions

Polynomial Expressions

b. –3y2 (y = –4) –3y2 –3(–4)2 = –3(16) = –48.

c. –3y3

–3y3 – 3(–4)3 = – 3(–64) = 192

The sum of monomials are called polynomials (many-terms), these are expressions of the form #xN ± #xN-1 ± … ± #x1 ± #where # can be any number.

For example, 4x + 7, –3x2 – 4x + 7, –5x4 + 1 are polynomials,

Polynomial Expressions

Polynomial Expressions

b. –3y2 (y = –4) –3y2 –3(–4)2 = –3(16) = –48.

c. –3y3

–3y3 – 3(–4)3 = – 3(–64) = 192

The sum of monomials are called polynomials (many-terms), these are expressions of the form #xN ± #xN-1 ± … ± #x1 ± #where # can be any number.

For example, 4x + 7, –3x2 – 4x + 7, –5x4 + 1 are polynomials,

x1 is not a polynomial.whereas the expression

Polynomial Expressions

Polynomial Expressions

Example C. Evaluate the polynomial 4x2 – 3x3 if x = –3.

Polynomial Expressions

Example C. Evaluate the polynomial 4x2 – 3x3 if x = –3.

The polynomial 4x2 – 3x3 is the combination of two

monomials; 4x2 and –3x3.

Polynomial Expressions

Example C. Evaluate the polynomial 4x2 – 3x3 if x = –3.

The polynomial 4x2 – 3x3 is the combination of two

monomials; 4x2 and –3x3. When evaluating the polynomial, we evaluate each monomial then combine the results.

Polynomial Expressions

Example C. Evaluate the polynomial 4x2 – 3x3 if x = –3.

The polynomial 4x2 – 3x3 is the combination of two

monomials; 4x2 and –3x3. When evaluating the polynomial, we evaluate each monomial then combine the results.Set x = (–3) in the expression,

Polynomial Expressions

Example C. Evaluate the polynomial 4x2 – 3x3 if x = –3.

The polynomial 4x2 – 3x3 is the combination of two

monomials; 4x2 and –3x3. When evaluating the polynomial, we evaluate each monomial then combine the results.Set x = (–3) in the expression, we get 4(–3)2 – 3(–3)3

Polynomial Expressions

Example C. Evaluate the polynomial 4x2 – 3x3 if x = –3.

The polynomial 4x2 – 3x3 is the combination of two

monomials; 4x2 and –3x3. When evaluating the polynomial, we evaluate each monomial then combine the results.Set x = (–3) in the expression, we get 4(–3)2 – 3(–3)3

= 4(9) – 3(–27)

Polynomial Expressions

Example C. Evaluate the polynomial 4x2 – 3x3 if x = –3.

The polynomial 4x2 – 3x3 is the combination of two

monomials; 4x2 and –3x3. When evaluating the polynomial, we evaluate each monomial then combine the results.Set x = (–3) in the expression, we get 4(–3)2 – 3(–3)3

= 4(9) – 3(–27)= 36 + 81 = 117

Polynomial Expressions

Example C. Evaluate the polynomial 4x2 – 3x3 if x = –3.

The polynomial 4x2 – 3x3 is the combination of two

monomials; 4x2 and –3x3. When evaluating the polynomial, we evaluate each monomial then combine the results.Set x = (–3) in the expression, we get 4(–3)2 – 3(–3)3

= 4(9) – 3(–27)= 36 + 81 = 117 Given a polynomial, each monomial is called a term.

Polynomial Expressions

Example C. Evaluate the polynomial 4x2 – 3x3 if x = –3.

The polynomial 4x2 – 3x3 is the combination of two

monomials; 4x2 and –3x3. When evaluating the polynomial, we evaluate each monomial then combine the results.Set x = (–3) in the expression, we get 4(–3)2 – 3(–3)3

= 4(9) – 3(–27)= 36 + 81 = 117 Given a polynomial, each monomial is called a term.

#xN ± #xN-1 ± … ± #x ± #

terms

Polynomial Expressions

Example C. Evaluate the polynomial 4x2 – 3x3 if x = –3.

The polynomial 4x2 – 3x3 is the combination of two

monomials; 4x2 and –3x3. When evaluating the polynomial, we evaluate each monomial then combine the results.Set x = (–3) in the expression, we get 4(–3)2 – 3(–3)3

= 4(9) – 3(–27)= 36 + 81 = 117 Given a polynomial, each monomial is called a term.

#xN ± #xN-1 ± … ± #x ± #

terms

Therefore the polynomial –3x2 – 4x + 7 has 3 terms, –3x2 , –4x and + 7.

Polynomial Expressions

Each term is addressed by the variable part.

Polynomial Expressions

Each term is addressed by the variable part. Hence the x2-term of the –3x2 – 4x + 7 is –3x2,

Polynomial Expressions

Each term is addressed by the variable part. Hence the x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x,

Polynomial Expressions

Each term is addressed by the variable part. Hence the x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x, and the number term or the constant term is 7.

Polynomial Expressions

Each term is addressed by the variable part. Hence the x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x, and the number term or the constant term is 7. The number in front of a term is called the coefficient of that term.

Polynomial Expressions

Each term is addressed by the variable part. Hence the x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x, and the number term or the constant term is 7. The number in front of a term is called the coefficient of that term. So the coefficient of –3x2 is –3 .

Polynomial Expressions

Each term is addressed by the variable part. Hence the x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x, and the number term or the constant term is 7. The number in front of a term is called the coefficient of that term. So the coefficient of –3x2 is –3 .

Operations with Polynomials

Polynomial Expressions

Each term is addressed by the variable part. Hence the x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x, and the number term or the constant term is 7. The number in front of a term is called the coefficient of that term. So the coefficient of –3x2 is –3 .

Terms with the same variable part are called like-terms. Operations with Polynomials

Polynomial Expressions

Each term is addressed by the variable part. Hence the x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x, and the number term or the constant term is 7. The number in front of a term is called the coefficient of that term. So the coefficient of –3x2 is –3 .

Terms with the same variable part are called like-terms. Like-terms may be combined.

Operations with Polynomials

Polynomial Expressions

Each term is addressed by the variable part. Hence the x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x, and the number term or the constant term is 7. The number in front of a term is called the coefficient of that term. So the coefficient of –3x2 is –3 .

Terms with the same variable part are called like-terms. Like-terms may be combined. For example, 4x + 5x = 9x

Operations with Polynomials

Polynomial Expressions

Each term is addressed by the variable part. Hence the x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x, and the number term or the constant term is 7. The number in front of a term is called the coefficient of that term. So the coefficient of –3x2 is –3 .

Terms with the same variable part are called like-terms. Like-terms may be combined. For example, 4x + 5x = 9x and 3x2 – 5x2 = –2x2.

Operations with Polynomials

Polynomial Expressions

Each term is addressed by the variable part. Hence the x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x, and the number term or the constant term is 7. The number in front of a term is called the coefficient of that term. So the coefficient of –3x2 is –3 .

Terms with the same variable part are called like-terms. Like-terms may be combined. For example, 4x + 5x = 9x and 3x2 – 5x2 = –2x2.Unlike terms may not be combined.

Operations with Polynomials

Polynomial Expressions

Each term is addressed by the variable part. Hence the x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x, and the number term or the constant term is 7. The number in front of a term is called the coefficient of that term. So the coefficient of –3x2 is –3 .

Terms with the same variable part are called like-terms. Like-terms may be combined. For example, 4x + 5x = 9x and 3x2 – 5x2 = –2x2.Unlike terms may not be combined. So x + x2 stays as x + x2.

Operations with Polynomials

Polynomial Expressions

Each term is addressed by the variable part. Hence the x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x, and the number term or the constant term is 7. The number in front of a term is called the coefficient of that term. So the coefficient of –3x2 is –3 .

Terms with the same variable part are called like-terms. Like-terms may be combined. For example, 4x + 5x = 9x and 3x2 – 5x2 = –2x2.Unlike terms may not be combined. So x + x2 stays as x + x2.Note that we write 1xN as xN , –1xN as –xN.

Operations with Polynomials

Polynomial Expressions

Each term is addressed by the variable part. Hence the x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x, and the number term or the constant term is 7. The number in front of a term is called the coefficient of that term. So the coefficient of –3x2 is –3 .

Terms with the same variable part are called like-terms. Like-terms may be combined. For example, 4x + 5x = 9x and 3x2 – 5x2 = –2x2.Unlike terms may not be combined. So x + x2 stays as x + x2.Note that we write 1xN as xN , –1xN as –xN.

When multiplying a number with a term, we multiply it with the coefficient.

Operations with Polynomials

Polynomial Expressions

Each term is addressed by the variable part. Hence the x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x, and the number term or the constant term is 7. The number in front of a term is called the coefficient of that term. So the coefficient of –3x2 is –3 .

Terms with the same variable part are called like-terms. Like-terms may be combined. For example, 4x + 5x = 9x and 3x2 – 5x2 = –2x2.Unlike terms may not be combined. So x + x2 stays as x + x2.Note that we write 1xN as xN , –1xN as –xN.

When multiplying a number with a term, we multiply it with the coefficient. Hence, 3(5x) = (3*5)x

Operations with Polynomials

Polynomial Expressions

Each term is addressed by the variable part. Hence the x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x, and the number term or the constant term is 7. The number in front of a term is called the coefficient of that term. So the coefficient of –3x2 is –3 .

Terms with the same variable part are called like-terms. Like-terms may be combined. For example, 4x + 5x = 9x and 3x2 – 5x2 = –2x2.Unlike terms may not be combined. So x + x2 stays as x + x2.Note that we write 1xN as xN , –1xN as –xN.

When multiplying a number with a term, we multiply it with the coefficient. Hence, 3(5x) = (3*5)x =15x,

Operations with Polynomials

Polynomial Expressions

Each term is addressed by the variable part. Hence the x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x, and the number term or the constant term is 7. The number in front of a term is called the coefficient of that term. So the coefficient of –3x2 is –3 .

Terms with the same variable part are called like-terms. Like-terms may be combined. For example, 4x + 5x = 9x and 3x2 – 5x2 = –2x2.Unlike terms may not be combined. So x + x2 stays as x + x2.Note that we write 1xN as xN , –1xN as –xN.

When multiplying a number with a term, we multiply it with the coefficient. Hence, 3(5x) = (3*5)x =15x, and –2(–4x) = (–2)(–4)x = 8x.

Operations with Polynomials

Polynomial Expressions

Each term is addressed by the variable part. Hence the x2-term of the –3x2 – 4x + 7 is –3x2, the x-term is –4x, and the number term or the constant term is 7. The number in front of a term is called the coefficient of that term. So the coefficient of –3x2 is –3 .

Terms with the same variable part are called like-terms. Like-terms may be combined. For example, 4x + 5x = 9x and 3x2 – 5x2 = –2x2.Unlike terms may not be combined. So x + x2 stays as x + x2.Note that we write 1xN as xN , –1xN as –xN.

When multiplying a number with a term, we multiply it with the coefficient. Hence, 3(5x) = (3*5)x =15x, and –2(–4x) = (–2)(–4)x = 8x.

Operations with Polynomials

When multiplying a number with a polynomial, we may expand using the distributive law: A(B ± C) = AB ± AC.

Polynomial Expressions

Example D. Expand and simplify.a. 3(2x – 4) + 2(4 – 5x)

Polynomial Expressions

Example D. Expand and simplify.a. 3(2x – 4) + 2(4 – 5x) = 6x – 12 + 8 – 10x

Polynomial Expressions

Example D. Expand and simplify.a. 3(2x – 4) + 2(4 – 5x) = 6x – 12 + 8 – 10x = –4x – 4

Polynomial Expressions

Example D. Expand and simplify.a. 3(2x – 4) + 2(4 – 5x) = 6x – 12 + 8 – 10x = –4x – 4 b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6)

Polynomial Expressions

Example D. Expand and simplify.a. 3(2x – 4) + 2(4 – 5x) = 6x – 12 + 8 – 10x = –4x – 4 b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6) = –3x2 + 9x – 15 + 2x2 + 8x +12

Polynomial Expressions

Example D. Expand and simplify.a. 3(2x – 4) + 2(4 – 5x) = 6x – 12 + 8 – 10x = –4x – 4 b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6) = –3x2 + 9x – 15 + 2x2 + 8x +12 = –x2 + 17x – 3

Polynomial Expressions

Example D. Expand and simplify.a. 3(2x – 4) + 2(4 – 5x) = 6x – 12 + 8 – 10x = –4x – 4

b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6) = –3x2 + 9x – 15 + 2x2 + 8x +12 = –x2 + 17x – 3

Polynomial Operations

When multiply a term with another term, we multiply the coefficient with the coefficient and the variable with the variable.

Example D. Expand and simplify.a. 3(2x – 4) + 2(4 – 5x) = 6x – 12 + 8 – 10x = –4x – 4

b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6) = –3x2 + 9x – 15 + 2x2 + 8x +12 = –x2 + 17x – 3

Polynomial Operations

When multiply a term with another term, we multiply the coefficient with the coefficient and the variable with the variable. Example E.

a. (3x2)(2x3) =

b. 3x2(–4x) =

c. 3x2(2x3 – 4x) =

Example D. Expand and simplify.a. 3(2x – 4) + 2(4 – 5x) = 6x – 12 + 8 – 10x = –4x – 4

b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6) = –3x2 + 9x – 15 + 2x2 + 8x +12 = –x2 + 17x – 3

Polynomial Operations

When multiply a term with another term, we multiply the coefficient with the coefficient and the variable with the variable. Example E.

a. (3x2)(2x3) = 3*2x2x3

b. 3x2(–4x) =

c. 3x2(2x3 – 4x) =

Example D. Expand and simplify.a. 3(2x – 4) + 2(4 – 5x) = 6x – 12 + 8 – 10x = –4x – 4

b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6) = –3x2 + 9x – 15 + 2x2 + 8x +12 = –x2 + 17x – 3

Polynomial Operations

When multiply a term with another term, we multiply the coefficient with the coefficient and the variable with the variable. Example E.

a. (3x2)(2x3) = 3*2x2x3 = 6x5

b. 3x2(–4x) =

c. 3x2(2x3 – 4x) =

Example D. Expand and simplify.a. 3(2x – 4) + 2(4 – 5x) = 6x – 12 + 8 – 10x = –4x – 4

b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6) = –3x2 + 9x – 15 + 2x2 + 8x +12 = –x2 + 17x – 3

Polynomial Operations

When multiply a term with another term, we multiply the coefficient with the coefficient and the variable with the variable. Example E.

a. (3x2)(2x3) = 3*2x2x3 = 6x5

b. 3x2(–4x) = 3(–4)x2x = –12x3

c. 3x2(2x3 – 4x) =

Example D. Expand and simplify.a. 3(2x – 4) + 2(4 – 5x) = 6x – 12 + 8 – 10x = –4x – 4

b. –3(x2 – 3x + 5) – 2(–x2 – 4x – 6) = –3x2 + 9x – 15 + 2x2 + 8x +12 = –x2 + 17x – 3

Polynomial Operations

When multiply a term with another term, we multiply the coefficient with the coefficient and the variable with the variable. Example E.

a. (3x2)(2x3) = 3*2x2x3 = 6x5

b. 3x2(–4x) = 3(–4)x2x = –12x3

c. 3x2(2x3 – 4x) distribute = 6x5 – 12x3

To multiply two polynomials, we may multiply each term of one polynomial against other polynomial then expand and simplify.

Polynomial Operations

To multiply two polynomials, we may multiply each term of one polynomial against other polynomial then expand and simplify.

Polynomial Operations

Example F.a. (3x + 2)(2x – 1)

To multiply two polynomials, we may multiply each term of one polynomial against other polynomial then expand and simplify.

Polynomial Operations

Example F.

= 3x(2x – 1) + 2(2x – 1)a. (3x + 2)(2x – 1)

To multiply two polynomials, we may multiply each term of one polynomial against other polynomial then expand and simplify.

Polynomial Operations

Example F.

= 3x(2x – 1) + 2(2x – 1)= 6x2 – 3x + 4x – 2

a. (3x + 2)(2x – 1)

To multiply two polynomials, we may multiply each term of one polynomial against other polynomial then expand and simplify.

Polynomial Operations

Example F.

= 3x(2x – 1) + 2(2x – 1)= 6x2 – 3x + 4x – 2= 6x2 + x – 2

a. (3x + 2)(2x – 1)

To multiply two polynomials, we may multiply each term of one polynomial against other polynomial then expand and simplify.

Polynomial Operations

Example F.

b. (2x – 1)(2x2 + 3x –4)

= 3x(2x – 1) + 2(2x – 1)= 6x2 – 3x + 4x – 2= 6x2 + x – 2

a. (3x + 2)(2x – 1)

To multiply two polynomials, we may multiply each term of one polynomial against other polynomial then expand and simplify.

Polynomial Operations

Example F.

b. (2x – 1)(2x2 + 3x –4)

= 3x(2x – 1) + 2(2x – 1)= 6x2 – 3x + 4x – 2= 6x2 + x – 2

= 2x(2x2 + 3x –4) –1(2x2 + 3x – 4)

a. (3x + 2)(2x – 1)

To multiply two polynomials, we may multiply each term of one polynomial against other polynomial then expand and simplify.

Polynomial Operations

Example F.

b. (2x – 1)(2x2 + 3x –4)

= 3x(2x – 1) + 2(2x – 1)= 6x2 – 3x + 4x – 2= 6x2 + x – 2

= 2x(2x2 + 3x –4) –1(2x2 + 3x – 4)= 4x3 + 6x2 – 8x – 2x2 – 3x + 4

a. (3x + 2)(2x – 1)

To multiply two polynomials, we may multiply each term of one polynomial against other polynomial then expand and simplify.

Polynomial Operations

Example F.

b. (2x – 1)(2x2 + 3x –4)

= 3x(2x – 1) + 2(2x – 1)= 6x2 – 3x + 4x – 2= 6x2 + x – 2

= 2x(2x2 + 3x –4) –1(2x2 + 3x – 4)= 4x3 + 6x2 – 8x – 2x2 – 3x + 4 = 4x3 + 4x2 – 11x + 4

a. (3x + 2)(2x – 1)

To multiply two polynomials, we may multiply each term of one polynomial against other polynomial then expand and simplify.

Polynomial Operations

Example F.

b. (2x – 1)(2x2 + 3x –4)

= 3x(2x – 1) + 2(2x – 1)= 6x2 – 3x + 4x – 2= 6x2 + x – 2

= 2x(2x2 + 3x –4) –1(2x2 + 3x – 4)= 4x3 + 6x2 – 8x – 2x2 – 3x + 4 = 4x3 + 4x2 – 11x + 4

a. (3x + 2)(2x – 1)

Note that if we did (2x – 1)(3x + 2) or (2x2 + 3x –4)(2x – 1) instead, we get the same answers. (Check this.)

To multiply two polynomials, we may multiply each term of one polynomial against other polynomial then expand and simplify.

Polynomial Operations

Example F.

b. (2x – 1)(2x2 + 3x –4)

= 3x(2x – 1) + 2(2x – 1)= 6x2 – 3x + 4x – 2= 6x2 + x – 2

= 2x(2x2 + 3x –4) –1(2x2 + 3x – 4)= 4x3 + 6x2 – 8x – 2x2 – 3x + 4 = 4x3 + 4x2 – 11x + 4

a. (3x + 2)(2x – 1)

Note that if we did (2x – 1)(3x + 2) or (2x2 + 3x –4)(2x – 1) instead, we get the same answers. (Check this.) Fact. If P and Q are two polynomials then PQ ≡ QP.

To multiply two polynomials, we may multiply each term of one polynomial against other polynomial then expand and simplify.

Polynomial Operations

Example F.

b. (2x – 1)(2x2 + 3x –4)

= 3x(2x – 1) + 2(2x – 1)= 6x2 – 3x + 4x – 2= 6x2 + x – 2

= 2x(2x2 + 3x –4) –1(2x2 + 3x – 4)= 4x3 + 6x2 – 8x – 2x2 – 3x + 4 = 4x3 + 4x2 – 11x + 4

a. (3x + 2)(2x – 1)

Note that if we did (2x – 1)(3x + 2) or (2x2 + 3x –4)(2x – 1) instead, we get the same answers. (Check this.) Fact. If P and Q are two polynomials then PQ ≡ QP. A shorter way to multiply is to bypass the 2nd step and use the general distributive law.

General Distributive Rule:Polynomial Operations

General Distributive Rule: (A ± B ± C ± ..)(a ± b ± c ..)

Polynomial Operations

General Distributive Rule: (A ± B ± C ± ..)(a ± b ± c ..)= Aa ± Ab ± Ac ..

Polynomial Operations

General Distributive Rule: (A ± B ± C ± ..)(a ± b ± c ..)= Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..

Polynomial Operations

General Distributive Rule: (A ± B ± C ± ..)(a ± b ± c ..)= Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc ..

Polynomial Operations

General Distributive Rule: (A ± B ± C ± ..)(a ± b ± c ..)= Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc ..

Example G. Expand

a. (x + 3)(x – 4)

Polynomial Operations

General Distributive Rule: (A ± B ± C ± ..)(a ± b ± c ..)= Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc ..

Example G. Expand

a. (x + 3)(x – 4)

= x2

Polynomial Operations

General Distributive Rule: (A ± B ± C ± ..)(a ± b ± c ..)= Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc ..

Example G. Expand

a. (x + 3)(x – 4)

= x2 – 4x

Polynomial Operations

General Distributive Rule: (A ± B ± C ± ..)(a ± b ± c ..)= Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc ..

Example G. Expand

a. (x + 3)(x – 4)

= x2 – 4x + 3x

Polynomial Operations

General Distributive Rule: (A ± B ± C ± ..)(a ± b ± c ..)= Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc ..

Example G. Expand

a. (x + 3)(x – 4)

= x2 – 4x + 3x – 12

Polynomial Operations

General Distributive Rule: (A ± B ± C ± ..)(a ± b ± c ..)= Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc ..

Example G. Expand

a. (x + 3)(x – 4)

= x2 – 4x + 3x – 12 simplify = x2 – x – 12

Polynomial Operations

General Distributive Rule: (A ± B ± C ± ..)(a ± b ± c ..)= Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc ..

Example G. Expand

a. (x + 3)(x – 4)

= x2 – 4x + 3x – 12 simplify = x2 – x – 12

b. (x – 3)(x2 – 2x – 2)

Polynomial Operations

General Distributive Rule: (A ± B ± C ± ..)(a ± b ± c ..)= Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc ..

Example G. Expand

a. (x + 3)(x – 4)

= x2 – 4x + 3x – 12 simplify = x2 – x – 12

b. (x – 3)(x2 – 2x – 2)

Polynomial Operations

= x3

General Distributive Rule: (A ± B ± C ± ..)(a ± b ± c ..)= Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc ..

Example G. Expand

a. (x + 3)(x – 4)

= x2 – 4x + 3x – 12 simplify = x2 – x – 12

b. (x – 3)(x2 – 2x – 2)

Polynomial Operations

= x3 – 2x2

General Distributive Rule: (A ± B ± C ± ..)(a ± b ± c ..)= Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc ..

Example G. Expand

a. (x + 3)(x – 4)

= x2 – 4x + 3x – 12 simplify = x2 – x – 12

b. (x – 3)(x2 – 2x – 2)

Polynomial Operations

= x3 – 2x2 – 2x

General Distributive Rule: (A ± B ± C ± ..)(a ± b ± c ..)= Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc ..

Example G. Expand

a. (x + 3)(x – 4)

= x2 – 4x + 3x – 12 simplify = x2 – x – 12

b. (x – 3)(x2 – 2x – 2)

Polynomial Operations

= x3 – 2x2 – 2x – 3x2

General Distributive Rule: (A ± B ± C ± ..)(a ± b ± c ..)= Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc ..

Example G. Expand

a. (x + 3)(x – 4)

= x2 – 4x + 3x – 12 simplify = x2 – x – 12

b. (x – 3)(x2 – 2x – 2)

Polynomial Operations

= x3 – 2x2 – 2x – 3x2 + 6x

General Distributive Rule: (A ± B ± C ± ..)(a ± b ± c ..)= Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc ..

Example G. Expand

a. (x + 3)(x – 4)

= x2 – 4x + 3x – 12 simplify = x2 – x – 12

b. (x – 3)(x2 – 2x – 2)

Polynomial Operations

= x3 – 2x2 – 2x – 3x2 + 6x + 6

General Distributive Rule: (A ± B ± C ± ..)(a ± b ± c ..)= Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc ..

Example G. Expand

a. (x + 3)(x – 4)

= x2 – 4x + 3x – 12 simplify = x2 – x – 12

b. (x – 3)(x2 – 2x – 2)

Polynomial Operations

= x3 – 2x2 – 2x – 3x2 + 6x + 6 = x3– 5x2 + 4x + 6

We will address the division operation of polynomials later-after we understand more about the multiplication operation.

General Distributive Rule: (A ± B ± C ± ..)(a ± b ± c ..)= Aa ± Ab ± Ac ..± Ba ± Bb ± Bc ..±Ca ± Cb ± Cc ..

Example G. Expand

a. (x + 3)(x – 4)

= x2 – 4x + 3x – 12 simplify = x2 – x – 12

b. (x – 3)(x2 – 2x – 2)

Polynomial Operations

= x3 – 2x2 – 2x – 3x2 + 6x + 6 = x3– 5x2 + 4x + 6

We will address the division operation of polynomials later-after we understand more about the multiplication operation.

Polynomials in two or more variables.

Polynomial Expressions

Polynomials in two or more variables. We form polynomials in two variables say, x & y, by adding monomials of the form kx#y# where k is a number and the powers are all nonnegative integers such as –5x3y2 or 3x2.

Polynomial Expressions

Polynomials in two or more variables. We form polynomials in two variables say, x & y, by adding monomials of the form kx#y# where k is a number and the powers are all nonnegative integers such as –5x3y2 or 3x2.Like–terms are terms where the variable parts are the same.For example 3x2y3 + 5x2y3 = 8x2y3 but 3x2y3 + 5x3y3 can’t be combined.

Polynomial Expressions

Polynomials in two or more variables. We form polynomials in two variables say, x & y, by adding monomials of the form kx#y# where k is a number and the powers are all nonnegative integers such as –5x3y2 or 3x2.Like–terms are terms where the variable parts are the same.For example 3x2y3 + 5x2y3 = 8x2y3 but 3x2y3 + 5x3y3 can’t be combined.

Polynomial Expressions

Example H. Expand and simplify.a. 2(3xy – 4x2y) + 2xy – 3xy2

Polynomials in two or more variables. We form polynomials in two variables say, x & y, by adding monomials of the form kx#y# where k is a number and the powers are all nonnegative integers such as –5x3y2 or 3x2.Like–terms are terms where the variable parts are the same.For example 3x2y3 + 5x2y3 = 8x2y3 but 3x2y3 + 5x3y3 can’t be combined.

Polynomial Expressions

Example H. Expand and simplify.a. 2(3xy – 4x2y) + 2xy – 3xy2

= 6xy – 8x2y + 2xy – 3xy2

Polynomials in two or more variables. We form polynomials in two variables say, x & y, by adding monomials of the form kx#y# where k is a number and the powers are all nonnegative integers such as –5x3y2 or 3x2.Like–terms are terms where the variable parts are the same.For example 3x2y3 + 5x2y3 = 8x2y3 but 3x2y3 + 5x3y3 can’t be combined.

Polynomial Expressions

Example H. Expand and simplify.a. 2(3xy – 4x2y) + 2xy – 3xy2

= 6xy – 8x2y + 2xy – 3xy2 = 8xy – 8x2y – 3xy2

Polynomials in two or more variables. We form polynomials in two variables say, x & y, by adding monomials of the form kx#y# where k is a number and the powers are all nonnegative integers such as –5x3y2 or 3x2.Like–terms are terms where the variable parts are the same.For example 3x2y3 + 5x2y3 = 8x2y3 but 3x2y3 + 5x3y3 can’t be combined. We evaluate them by assigning numbers to x and/or y.

Polynomial Expressions

Example H. Expand and simplify.a. 2(3xy – 4x2y) + 2xy – 3xy2

= 6xy – 8x2y + 2xy – 3xy2 = 8xy – 8x2y – 3xy2

Polynomials in two or more variables. We form polynomials in two variables say, x & y, by adding monomials of the form kx#y# where k is a number and the powers are all nonnegative integers such as –5x3y2 or 3x2.Like–terms are terms where the variable parts are the same.For example 3x2y3 + 5x2y3 = 8x2y3 but 3x2y3 + 5x3y3 can’t be combined. We evaluate them by assigning numbers to x and/or y.

Polynomial Expressions

Example H. Expand and simplify.a. 2(3xy – 4x2y) + 2xy – 3xy2

= 6xy – 8x2y + 2xy – 3xy2 = 8xy – 8x2y – 3xy2

b. Evaluate 8xy – 8x2y – 3xy2 if x = 2.

Polynomials in two or more variables. We form polynomials in two variables say, x & y, by adding monomials of the form kx#y# where k is a number and the powers are all nonnegative integers such as –5x3y2 or 3x2.Like–terms are terms where the variable parts are the same.For example 3x2y3 + 5x2y3 = 8x2y3 but 3x2y3 + 5x3y3 can’t be combined. We evaluate them by assigning numbers to x and/or y. If only one number is given, the result is a formula.

Polynomial Expressions

Example H. Expand and simplify.a. 2(3xy – 4x2y) + 2xy – 3xy2

= 6xy – 8x2y + 2xy – 3xy2 = 8xy – 8x2y – 3xy2

b. Evaluate 8xy – 8x2y – 3xy2 if x = 2.

Polynomials in two or more variables. We form polynomials in two variables say, x & y, by adding monomials of the form kx#y# where k is a number and the powers are all nonnegative integers such as –5x3y2 or 3x2.Like–terms are terms where the variable parts are the same.For example 3x2y3 + 5x2y3 = 8x2y3 but 3x2y3 + 5x3y3 can’t be combined. We evaluate them by assigning numbers to x and/or y. If only one number is given, the result is a formula.

Polynomial Expressions

Example H. Expand and simplify.a. 2(3xy – 4x2y) + 2xy – 3xy2

= 6xy – 8x2y + 2xy – 3xy2 = 8xy – 8x2y – 3xy2

b. Evaluate 8xy – 8x2y – 3xy2 if x = 2.Input x = 2, we get 8(2)y – 8(2)2y – 3(2)y2

Polynomials in two or more variables. We form polynomials in two variables say, x & y, by adding monomials of the form kx#y# where k is a number and the powers are all nonnegative integers such as –5x3y2 or 3x2.Like–terms are terms where the variable parts are the same.For example 3x2y3 + 5x2y3 = 8x2y3 but 3x2y3 + 5x3y3 can’t be combined. We evaluate them by assigning numbers to x and/or y. If only one number is given, the result is a formula.

Polynomial Expressions

Example H. Expand and simplify.a. 2(3xy – 4x2y) + 2xy – 3xy2

= 6xy – 8x2y + 2xy – 3xy2 = 8xy – 8x2y – 3xy2

b. Evaluate 8xy – 8x2y – 3xy2 if x = 2.Input x = 2, we get 8(2)y – 8(2)2y – 3(2)y2

= 16y – 32y – 6y2 = –16y – 6y2

Polynomials in two or more variables. We form polynomials in two variables say, x & y, by adding monomials of the form kx#y# where k is a number and the powers are all nonnegative integers such as –5x3y2 or 3x2.Like–terms are terms where the variable parts are the same.For example 3x2y3 + 5x2y3 = 8x2y3 but 3x2y3 + 5x3y3 can’t be combined. We evaluate them by assigning numbers to x and/or y. If only one number is given, the result is a formula. If both numbers are given, then we get a numerical output.We may do this for x, y and z or even more variables.

Polynomial Expressions

Example H. Expand and simplify.a. 2(3xy – 4x2y) + 2xy – 3xy2

= 6xy – 8x2y + 2xy – 3xy2 = 8xy – 8x2y – 3xy2

b. Evaluate 8xy – 8x2y – 3xy2 if x = 2.Input x = 2, we get 8(2)y – 8(2)2y – 3(2)y2

= 16y – 32y – 6y2 = –16y – 6y2

Ex. A. Evaluate each monomials with the given values.

3. 2x2 with x = 1 and x = –1

4. –2x2 with x = 1 and x = –1

5. 5y3 with y = 2 and y = –2

6. –5y3 with y = 2 and y = –2

1. 2x with x = 1 and x = –1

2. –2x with x = 1 and x = –1

7. 5z4 with z = 2 and z = –2

8. –5y4 with z = 2 and z = –2

B. Evaluate each monomials with the given values.9. 2x2 – 3x + 2 with x = 1 and x = –1

10. –2x2 + 4x – 1 with x = 2 and x = –2

11. 3x2 – x – 2 with x = 3 and x = –3

12. –3x2 – x + 2 with x = 3 and x = –3

13. –2x3 – x2 + 4 with x = 2 and x = –2

14. –2x3 – 5x2 – 5 with x = 3 and x = –3

C. Expand and simplify.15. 5(2x – 4) + 3(4 – 5x) 16. 5(2x – 4) – 3(4 – 5x)

17. –2(3x – 8) + 3(4 – 9x) 18. –2(3x – 8) – 3(4 – 9x)

19. 7(–2x – 7) – 3(4 – 3x) 20. –5(–2 – 8x) + 7(–2 – 11x)

Polynomial Expressions

21. x2 – 3x + 5 + 2(–x2 – 4x – 6)22. x2 – 3x + 5 – 2(–x2 – 4x – 6)23. 2(x2 – 3x + 5) + 5(–x2 – 4x – 6)24. 2(x2 – 3x + 5) – 5(–x2 – 4x – 6)25. –2(3x2 – 2x + 5) + 5(–4x2 – 4x – 3)26. –2(3x2 – 2x + 5) – 5(–4x2 – 4x – 3)27. 4(3x3 – 5x2) – 9(6x2 – 7x) – 5(– 8x – 2)

29. Simplify 2(3xy – xy2) – 2(2xy – xy2) then evaluated it with x = –1, afterwards evaluate it at (–1, 2) for (x, y) 30. Simplify x2 – 2(3xy – x2) – 2(y2 – xy) then evaluated it with y = –2, afterwards evaluate it at (–1, –2) for (x, y) 31. Simplify x2 – 2(3xy – z2) – 2(z2 – x2) then evaluated it with x = –1, y = – 2 and z = 3.

Polynomial Expressions

28. –6(7x2 + 5x – 9) – 7(–3x2 – 2x – 7)