Essential Question: How can you determine if x-2 is a factor of a polynomial without

factoring?

Standard Form◦ Largest Exponent comes first.◦ Combine like terms (if possible)◦ The constant (number without a variable) comes

last Example

◦ Write the following polynomials in standard form -7x + 5x4

x2 – 4x + 3x3 + 2x

5x4 – 7x

3x3 + x2 – 2x

Degree◦ The degree of a polynomial is the largest exponent

Example◦ Find the degree of the polynomial

5x4 - 7x

Quartic Cubic

Degree

Name Using Degree

Example

0 Constant 6

1 Linear x + 3

2 Quadratic 3x2

3 Cubic 2x3 – 5x2 – 2x

4 Quartic x4 + 3x2

5 Quintic -2x5 + 3x2 – x + 4

• 3x3 + x2 - 2x

Number of Terms

Example◦ Classify each polynomial by the number of terms

5x4 - 7x

Binomial Trinomial

Number of Terms

Name using Number of Terms

Example

1 Monomial 6

2 Binomial x + 3

3 Trinomial 2x3 – 5x2 – 2x

More than 3 Polynomial of x terms

-2x5 + 3x2 – x + 4

• 3x3 + x2 - 2x

Write the expression (x + 1)(x + 2)(x + 3) as a polynomial in standard form.◦ FOIL the last two terms

(x + 1)(x + 2)(x + 3) (x + 1)(x2 + 3x + 2x + 6) (x + 1)(x2 + 5x + 6)

◦ Distribute the (x + 1) to all terms

(x + 1)(x2 + 5x + 6)

x3 + 5x2 + 6x + x2 + 5x + 6◦ Combine like terms

x3 + 6x2 + 11x + 6

YOUR TURNWrite the expression (x + 1)(x + 1)(x + 2) as a polynomial in standard form.

(x + 1)(x + 1)(x + 2)(x + 1)(x2 + 2x + 1x + 2)(x + 1)(x2 + 3x + 2)x3 + 3x2 + 2x + x2 + 3x + 2x3 + 4x2 + 5x + 2

Writing a Polynomial in Factored Form◦ Write 2x3 + 10x2 + 12x in factored form◦ Factor out a GCF first

2x3 + 10x2 + 12x 2x(x2 + 5x + 6)

◦ Factor the quadratic in parenthesis 2x(x + 2)(x + 3)

YOUR TURNWrite 3x3 – 3x2 – 36x in factored form

3x(x2 – x – 12)3x(x – 4)(x + 3)

Assignment◦ Page 309

Problems 1 – 11 (odd problems) Make sure to include the original problem

◦ Page 317 Problems 1 – 6 (all problems) Problems 7 – 11 (odd problems) Show your work

Remember: (x – 3)2 means (x – 3)(x – 3), not x2 + 9

Essential Question: How can you determine if x-2 is a factor of a polynomial without

factoring?

If a polynomial is in factored form, you can use the Zero Product Property to find values that will make the polynomial equal zero.

Example◦ Find the zeros of y = (x – 2)(x + 1)(x + 3).◦ Just like factoring, if any of the parenthesis come out

as zero, then the function is zero. x – 2 = 0 or x + 1 = 0 or x + 3 = 0

x + 2 = +2 x – 1 = -1 x – 3 = -3 -2 x = 2 +1 x = -1 +3 x = -3

Your Turn◦ Find the zeros of the function y = (x – 7)(x – 5)(x – 3)◦ x = 7, 5, or 3

Writing a Polynomial Function From Its Zeros◦ Write a polynomial function in standard form with

zeros at -2, 3 and 3◦ Just the opposite of what we did in the last

example, except we also have to multiply the factors together (x + 2)(x – 3)(x – 3)

FOIL the last two terms (x + 2)(x2 – 6x + 9)

Distribute the x + 2 to all terms x3 – 6x2 + 9x + 2x2 – 12x + 18

Combine like terms x3 – 4x2 – 3x + 18

YOUR TURNWrite a polynomial function in standard form with zeros at -4, -2 and 1

(x + 4)(x + 2)(x - 1)(x + 4)(x2 – 1x + 2x - 2)(x + 4)(x2 + x - 2)x3 + x2 – 2x + 4x2 + 4x - 8x3 + 5x2 + 2x – 8

Multiplicity◦ Sometimes, a zero can show up multiple times.

Though we generally don’t list multiple zeros as solutions, a multiple zero has MULTIPLICITY equal to the number of times the zero occurs.

◦ Example: f(x) = x4 + 6x3 + 8x2

f(x) = x2(x2 + 6x + 8) f(x) = x2(x + 4)(x + 2)

Note: you can rewrite x2 as (x – 0)2 or (x – 0)(x – 0) The zeros are x = 0 (multiplicity 2), x = -4, and x = -

2

Find the zeros of the function. State any multiplicity of multiple zeros.◦ f(x) = (x – 2)(x - 1)(x + 1)2

x = 2, x = 1, x = -1 (multiplicity 2)

◦ y = x3 – 4x2 + 4x y = x(x2 – 4x + 4) y = x(x – 2)(x – 2) x = 0, x = 2 (multiplicity 2)

Assignment◦ Page 317 - 318

Problems 16 – 27 (all problems) Problems 29 – 35, 41 – 45 (odd problems)

Show your work (not relevant in 16-20) Don’t graph problems 16 - 20