2-1 Operations on Polynomials. Refer to the algebraic expression above to complete the following: 1)How many terms are there? 2)Give an example of like.

Secondary Math 32-1 Operations on Polynomials

4424 15672 xxxx

Refer to the algebraic expression above to complete the following:1) How many terms are there?2) Give an example of like terms.3) Give an example of a coefficient.4) Give an example of a constant.5) Simplify the expression.

4424 15672 xxxx 1) There are 6 terms – a term is a constant or a

variable or a product of a constant and a variable separated by + and – signs.

2) -6, 1 and are like terms – terms with the same variable to the same power.

3) 2, 7, -5, and -1 are coefficients – when the term contains a number and a variable, the number part is the coefficient.

4) -6 and 1 are constants – a term that does not have a variable.

444 ,5,2 xxx

4424 15672 xxxx

To simplify the expression:Add the coefficients of any like terms.Keep the variable and exponent the same.Write in order with the largest exponent first which is the standard (general form) of a polynomial.

574 24 xx

An expression formed by adding a finite number of “same base” unlike terms.

Example:

Exponents must be positive integers (no fractions), there can be no square roots, and no variables in the denominator.

- Not a polynomial

Polynomial -

164 23 xx

52 13

2

xx

The exponent of a term is the degree of the term.

Example has a degree of 5

The value of the largest term is the degree of a polynomial.

Example has a degree of 3

The leading coefficient is the coefficient of the first term when the polynomial is written in standard form (largest degree first).

Degree -

59x

164 23 xx

Your turn – Write each polynomial in standard form, name the degree and the leading coefficient.

42 64 yy 1)

2) 9 + 3x

3) 734 23 zz

Naming Polynomials

Naming Polynomials

Naming Polynomials

Naming Polynomials

Naming Polynomials

Adding Polynomials

)15()23( 4545 xxxxDrop parentheses and add like terms.

Make sure answer is in standard form

Adding Polynomials - Vertically

)15()23( 4545 xxxx

Line up terms by degree

15

2345

45

xx

xx

Your Turn - simplify

)65()65( 232 xxxx

Subtracting Polynomials

)753()283( 22 xxxx

Change the signs of the second polynomial and then add.

Subtracting - Vertically

)753()283( 22 xxxx

283 2 xx)753( 2 xx

Your turn - Simplify

)8()145( 22 xxx

If you add or subtract polynomials your answer is also a polynomial.

This means polynomials are “closed” under addition and subtraction.

Multiplying Polynomials

2(3x – 1) Distributive property

)75(2 23 xxx

Your Turn - simplify

34 1410 xx

If you multiply polynomials you get a polynomial as the answer.

Polynomials are “closed” for multiplication!!!

Division of Polynomials

2x

x1x

Let’s do a very simple one.

Polynomials are NOT “closed” for divison!!!

Multiplying Polynomials

(x + 3)(2x – 1) Distributive property (twice)

)12( xx )12(3 x

22x x x6 3 Combine “like” terms

352 2 xx

)4)(52( xx

Your Turn: simplify

)3)(4( xx 122 xx

2032 2 xx

)42( 2 xxx

Distributive property (twice)

)42(3 2 xx3x 22x x4 23x

Combine “like” terms3x

)42)(3( 2 xxx

x6 12

2x x10

Multiplying PolynomialsHow do you multiply 2 * 3 * 4 three numbers?

6 * 4 = 24OR: 2 * 3 * 4

122 * = 24 OR: 2 * 3 * 4

8 * 3 = 24

Pick 2 factors, multiply them to get a product, then multiply the product by the last factor

New Property

Associative Property: if you have 3 or more factors, pick two, multiply them 1st

2*3*4

(2*3)*4

(to visually show that we are picking 2, we group, or associate them together with parentheses).

Multiplying Polynomials

(x – 1)(2x + 3)(3x – 2) = ?

= [ (x – 1)(2x + 3) ] (3x – 2)

)23)(3322( 2 xxxx

)23)(32( 2 xxx

How do you multiply three binomials? Pick 2 factors, multiply them to get a product, then multiply the product by the last factor associative property.

)23()32(1)32( xxxx

Multiplying Polynomials Vertically

?)23)(32( 2 xxx

)32( 2 xx

)23( x36x 23x x9

Special ProductsSquare of a sum.

(x + y)(x + y)

2)( yx

Special ProductsSquare of a difference.

(x - y)(x - y)

2)( yx

Special ProductsProduct of a sum and a difference.

(x + y)(x – y)

)1)(1( 22 xx

Your Turn: Multiply

)34)(24( 23 xx

3)1( x

3)( yx

Special ProductsCube of a sum.

))(2( 22 yxyxyx )()( 2 yxyx

3223 33 yxyyxx

3)2( x 3223 )2()2)((3)2()(3)( xxx

3223 ) () )( (3) () (3) (

8126 23 xxx

3)( yx

Special ProductsCube of a difference.

))(2( 22 yxyxyx )()( 2 yxyx

3223 33 yxyyxx 3223 ) () )( (3) () (3) (

3)2( x 3223 )2()2)(x(3)2()x(3)x(

8126 23 xxx

3)4( x

Your Turn: Multiply

3)32( x