5.3: Add, Subtract, & Multiply 5.3: Add, Subtract, & Multiply PolynomialsPolynomials

Objectives:

1. To define a polynomial and related terms

2. To find the sum, difference, and product of polynomials

PolynomialPolynomial

What makes one of these things a polynomial?

Polynomial NOT a Polynomial

2x2 3xy

x4 – 81 2x - √x

x2 – 3x + 4 x-1 + 2x-2 – 4x-3

PolynomialPolynomial

A polynomialpolynomial in x is an expression of the form:

2 02

1nnx xa a a ax

Monomial Binomial Trinomial

One term Two terms Three terms

2x2 x4 – 81 x2 – 3x + 4

Like TermsLike Terms

Like terms Like terms are simply monomials with the same variable raised to the same power

• To add and subtract polynomials, just add or subtract the coefficients of like terms– The powers DO NOT change!

3 2 3 22 4 5 7x x x x x

Exercise 1Exercise 1

Add using a horizontal format.

It’s often helpful to underline like terms the same number of times.

3 2 3 24 4 3 10 5 2 4 4x x x x x x

3x 22x 7x 6

Exercise 2Exercise 2

Add using a vertical format.

Align like terms and add the old-fashioned way.

3 2 3 22 2 3 5 3 4 7x x x x x x

3 2

3 2

2 2 3 5

3 4 7

x x x

x x x

35x 22x 4x 2

Distributive PropertyDistributive Property

Distributive PropertyDistributive Property

When subtracting polynomials, you have to distribute the negative sign:

3 23 4 7x x x 3 21 3 4 7x x x 33x 24x x 7

Exercise 3Exercise 3

Subtract 6y2 – 6y – 13 from 3y2 – 4y + 7 in a horizontal format.

2 23 4 7 6 6 13y y y y 2 23 4 7 6 6 13y y y y

23y 2y 20

Exercise 4Exercise 4

Subtract –4x3 + 6x2 + 9x – 3 from 3x3 + 4x2 + 7x + 12 in a vertical format.

Basically you have to turn a subtraction problem into addition by adding the opposite.

3 2

3 2

3 4 7 12

( 4 6 9 3)

x x x

x x x

3 2

3 2

3 4 7 12

4 6 9 3

x x x

x x x

37x 22x 2x 15

Exercise 5Exercise 5

Find the sum or difference.

1.(t2 – 6t + 2) + (5t2 – t – 8)

2.(8d – 3 + 9d3) – (d3 – 13d2 – 4)

Polynomial MultiplicationPolynomial Multiplication

Polynomial multiplication is basically repeated application of the distributive property.

Multiply coefficients and add exponents

26 3 4 1x x x 26 3x x 6 4x x 6 1x

318x 224x 6x

Polynomial MultiplicationPolynomial Multiplication

Polynomial multiplication is basically repeated application of the distributive property.

2 3 5 1x x 2 5 1x x 3 5 1x 210x 2x 15x 3

Product of the FirstFirst terms

Product of the FirstFirst terms

Product of the OutsideOutside termsProduct of the OutsideOutside terms

Product of the Inside Inside terms

Product of the Inside Inside terms

Product of the Last Last terms

Product of the Last Last terms

210 13 3x x

Polynomial MultiplicationPolynomial Multiplication

Polynomial multiplication is basically repeated application of the distributive property.

2 3 5 1x x 210x 2x 15x 3210 13 3x x

FFirstOOutsideIInsideLLast

Exercise 6Exercise 6

Find the product.

Difference of two squares

a b a b 2 2a b

Protip: Difference of 2 Protip: Difference of 2 SquaresSquaresWhen finding the difference of 2 squares,

just square the first number, square the second number, and take the difference. The middle term cancels out.

a b a b 2 2a b Square the first term

Square the second term

Exercise 8Exercise 8

1. (5y – 3)(5y + 3)

2. (4a + 7)2

3. (2x – 3)2

Polynomial MultiplicationPolynomial Multiplication

When multiplying a polynomial by a polynomial, each term of the first polynomial must be multiplied by each term of the second polynomial.

• Again, this is just the distributive property used multiple times.

Exercise 9Exercise 9

Multiply x2 – 2x + 3 and x + 5 in a horizontal format.

25 2 3x x x 25x x 5 2x x 5 3x

3x 25x 22x 10x 3x 153 23 7 15x x x

Exercise 9Exercise 9

Multiply x2 – 2x + 3 and x + 5 in a horizontal format.

25 2 3x x x 3x 25x22x 10x3x 153 23 7 15x x x

Exercise 10Exercise 10

Multiply 3x2 + 3x + 5 and 2x + 3 in a vertical format.

23 3 5

2 3

x x

x

159x29x

Exercise 10Exercise 10

Multiply 3x2 + 3x + 5 and 2x + 3 in a vertical format.

23 3 5

2 3

x x

x

159x29x

10x26x36x36x 215x 19x 15

Exercise 11Exercise 11

Find the product.

1.(x + 2)(3x2 – x – 5)

2.(x – 3)(4 + 2x – x2)

Exercise 14Exercise 14

Find the product.

(a – 5)(a + 2)(a + 6)