U2 – 2.1 P OLYNOMIALS Naming Polynomials Add and Subtract Polynomials Multiply Polynomials.

U2 – 2.1

POLYNOMIALS

Naming PolynomialsAdd and Subtract PolynomialsMultiply Polynomials

DEFINITIONSExponentsPowerSimplifyTermsMonomialsPolynomials

MORE DEFINITIONSLike TermsConstantDegreeCoefficientBinomialTrinomial

EXAMPLE 1 Identify polynomial functions

Name each polynomial by degree (highest exponent), the number of terms (and expression that can be written as a sum, the parts added together) leading coefficient (number in front of the variable with highest degree)

a. 2 + 5

SOLUTION

a. The degree is 4, number of terms is 2, leading coefficient 2

EXAMPLE 1 Identify polynomial functions

Name each polynomial by degree, the number of terms, and leading coefficient.

a. 10a

SOLUTION

a. The degree is 1, the number of terms is 1, and the leading coefficient is 10.

EXAMPLE 1 Identify polynomial functions

Name each polynomial by degree, the number of terms, leading coefficient

a. - 5 +10n - 2

SOLUTION

a. The degree is 5, the number of terms is 3, and the leading coefficient is 3.

EXAMPLE 1 Identify polynomial functions

Name each polynomial by degree, the number of terms, and leading coeffcient.

a. 3

SOLUTION

a. The degree is :None and the number of terms is 1, leading coefficient none.

GUIDED PRACTICE for Examples 1 and 2

State the polynomials degree, terms, and leading coefficient.

1. f (x) = 13 – 2x

SOLUTION

f (x) = – 2x + 13It is a polynomial function.Standard form: – 2x + 13Degree: 1 Leading coefficient of – 2.Number of terms : 2

GUIDED PRACTICE for Examples 1 and 2

2. p (x) = 9x4 – 5x 2 + 4

SOLUTION

It is a polynomial function.Standard form: . p (x) = 9x4 – 5x 2 +

4Degree: 4 Leading coefficient of 9.Number of terms : 4

GUIDED PRACTICE for Examples 1 and 2

3. h (x) = 6x2 + π – 3x

SOLUTION h (x) = 6x2 – 3x + π

The function is a polynomial function that is already written in standard form will be 6x2– 3x + π .

It has degree 2 and a leading coefficient of 6.

It is a polynomial function.Standard form: 6– 3x + π Degree: 2 Terms: 3Leading coefficient of 6

EXAMPLE 1 Add polynomials vertically and horizontally

a. Add 2x3 – 5x2 + 3x – 9 and x3 + 6x2 + 11 in a vertical

format.

SOLUTION

a. 2x3 – 5x2 + 3x – 9

+ x3 + 6x2 + 11

3x3 + x2 + 3x + 2

EXAMPLE 1 Add polynomials vertically and horizontally

(3y3 – 2y2 – 7y) + (– 4y2 + 2y – 5)

= 3y3 – 2y2 – 4y2 – 7y + 2y – 5

= 3y3 – 6y2 – 5y – 5

b. Add 3y3 – 2y2 – 7y and – 4y2 + 2y – 5 in a horizontal format.

EXAMPLE 2 Subtract polynomials vertically and horizontally

a. Subtract 3x3 + 2x2 – x + 7 from 8x3 – x2 – 5x + 1 in a vertical format.

SOLUTION

a. Align like terms, then add the opposite of the subtracted polynomial.

8x3 – x2 – 5x + 1

– (3x3 + 2x2 – x + 7)

8x3 – x2 – 5x + 1

+ – 3x3 – 2x2 + x – 7

5x3 – 3x2 – 4x – 6

EXAMPLE 2

Write the opposite of the subtracted polynomial, then add like terms.

(4z2 + 9z – 12) – (5z2 – z + 3) = 4z2 + 9z – 12 – 5z2 + z – 3

= 4z2 – 5z2 + 9z + z – 12 – 3

= – z2 + 10z – 15

Subtract polynomials vertically and horizontally

b. Subtract 5z2 – z + 3 from 4z2 + 9z – 12 in a horizontal

format.

GUIDED PRACTICE for Examples 1 and 2

Find the sum or difference.

1. (t2 – 6t + 2) + (5t2 – t – 8)

SOLUTION

6t2 – 7t – 6

t2 – 6t + 2

+ 5t2 – t – 8

GUIDED PRACTICE for Examples 1 and 2

2. (8d – 3 + 9d3) – (d3 – 13d2 – 4)

SOLUTION

= (8d – 3 + 9d3) – (d3 – 13d2 – 4)

= (8d – 3 + 9d3) – d3 + 13d2 + 4)

= 9d3 –3 d3 + 13d2 + 8d – 3 + 4

= 8d3 + 13d2 + 8d + 1

TRY THE FOLLOWING PROBLEMS

1.( 3 - 5 ) + ( 7 - 3 )

2.( 5 + 7c + 3 ) + ( + 6c + 4 )

3. ( - 3 - 2x + 2) + ( + 6 + 4 )

ANSWERS TO ADDITION PROBLEMS

1. ( 3 - 5 ) + ( 7 - 3 ) = 10 - 8

2. ( 5 + 7c + 3 ) + ( + 6c + 4 ) = 6 + 13c + 7

3. ( - 3 - 2x + 2) + ( + 6 + 4 ) = 11 -2x + 6

TRY THE FOLLOWING PROBLEMS

1.( 4 - 9 ) - ( 2 - 3 )

2.( 4 + 2c + 6 ) - ( + 8c + 4 )

3. ( - 8 - 3x + 9 ) - ( - + 4 + 5 )

ANSWERS TO THE SUBTRACT PROBLEMS

1. ( 4 - 9 ) - ( 2 - 3 ) = - 6

2. ( 4 + 2c + 6 ) - ( + 8c + 4 ) = 3

3. ( - 8 - 3x + 9 ) - ( - + 4 + 5 )

= -3x + 4

Multiply Polynomials

1.3(2) 2.3(2x)

3.3x(2x) 4.3x(2x+1)

5.3x+1(2x) 6.(3x+1)(2x+1)

7.(3x+1)(2x+1)(x+1)