Do Now:

• Pass out calculators.1. Compare and contrast factoring:

6x2 – x – 2 with factoring x2 – x – 2 • Factor both of the problems above. • Write a few sentences explaining the

similarities and differences about the process of factoring each.

Do Now:

• Pass out calculators.

Objective:

• To factor polynomials completely.

Factor out a common binomialEXAMPLE 1

2x(x + 4) – 3(x + 4)a.

SOLUTION

3y2(y – 2) + 5(2 – y)b.

2x(x + 4) – 3(x + 4) = (x + 4)(2x – 3)a.

The binomials y – 2 and 2 – y are opposites. Factor – 1 from 2 – y to obtain a common binomial factor.

b.

3y2(y – 2) + 5(2 – y) = 3y2(y – 2) – 5(y – 2)

= (y – 2)(3y2 – 5)

Factor – 1 from (2 – y).

Distributive property

Factor the expression.

Factor by groupingEXAMPLE 2

x3 + 3x2 + 5x + 15.a y2 + y + yx + xb.

SOLUTION

= x2(x + 3) + 5(x + 3)= (x + 3)(x2 + 5)

x3 + 3x2 + 5x + 15 = (x3 + 3x2) + (5x + 15)a.

y2 + y + yx + x = (y2 + y) + (yx + x)b.= y(y + 1) + x(y + 1)= (y + 1)(y + x)

Group terms.

Factor each group.Distributive property

Group terms.

Factor each group.

Distributive property

Factor the polynomial.

Factor by groupingEXAMPLE 3

Factor 6 + 2x .x3 – 3x2–

SOLUTION

The terms x2 and –6 have no common factor. Use the commutative property to rearrange the terms so that you can group terms with a common factor.

x3 – 3x2 + 2x – 6 x3 – 6 + 2x – 3x2 = Rearrange terms.

(x3 – 3x2 ) + (2x – 6)= Group terms.

x2 (x – 3 ) + 2(x – 3) = Factor each group.

(x – 3)(x2+ 2) = Distributive property

Factor by grouping

EXAMPLE 3

CHECK

Check your factorization using a graphing calculator. Graph

y and

y

Because the graphs coincide, you know that your factorization is correct.

1

= (x – 3)(x2 + 2) .2

6 + 2x = x3 – – 3x2

GUIDED PRACTICE for Examples 1, 2 and 3

Factor the expression.

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

2. a3 + 3a2 + a + 3 = (a + 3)(a2 + 1)

3. y2 + 2x + yx + 2y = (y + 2)( y + x )

Do Now:

• Pass out calculators. • Pick up a homework answer key

from the table and make corrections to your homework.

To Factor COMPLETELY…1. Factor out greatest common monomial factor (if

possible). Example: 3x2 + 6x = 3x (x + 2)

2. Look for a difference of two squares or a perfect square trinomial.

Example: x2 + 4x + 4 = (x + 2)2

3. Factor the trinomial into a product of factors. Example: 3x2 – 5x – 2 = (3x + 1)(x – 2)

4. Factor a polynomial with four terms by grouping. Example: x3 + x – 4x2 – 4 = (x2 + 1)(x – 4)

To Factor COMPLETELY…

• A polynomial is factored completely when there is no other possible way to factor it.

• It should be written as a product of unfactorable polynomials with integer coefficients.

Factor completely

EXAMPLE 1

Factor the polynomial completely.

a. n2 + 2n – 1

SOLUTION

a. The terms of the polynomial have no common monomial factor. Also, there are no factors of – 1 that have a sum of 2. This polynomial cannot be factored.

Factor completely

EXAMPLE 2

Factor the polynomial completely.

b. 4x3 – 44x2 + 96x

SOLUTION

b. 4x3 – 44x2 + 96x = 4x(x2– 11x + 24) Factor out 4x.

= 4x(x– 3)(x – 8)Find two negativefactors of 24 that

have a sum of – 11.

Factor completely

EXAMPLE 2

Factor the polynomial completely.

c. 50h4 – 2h2

SOLUTION

c. 50h4 – 2h2 = 2h2 (25h2 – 1) Factor out 2h2.

= 2h2 (5h – 1)(5h + 1) Difference of two squares pattern

GUIDED PRACTICE for Example 4

Factor the polynomial completely.1. 3x3 – 12x = 3x (x + 2)(x – 2)

2. 2y3 – 12y2 + 18y = 2y(y – 3)2

3. m3 – 2m2 + 8m = m(m – 4)(m + 2)

Solve a polynomial equation

EXAMPLE 3

Factor out 3x.

Solve 3x3 + 18x2 = – 24x .

3x3 + 18x2 = – 24x Write original equation.

3x3 + 18x2 + 24x = 0 Add 24x to each side.

3x(x2 + 6x + 8) = 0

3x(x + 2)(x + 4) = 0 Factor trinomial.

Zero-product property

x = 0 or x = – 2 or x = – 4 Solve for x.

3x = 0 or x + 2 = 0 or x + 4 = 0

ANSWER

The solutions of the equation are 0, – 2, and – 4.

GUIDED PRACTICE for Example 5

Solve the equation.

4. w3 – 8w2 + 16w = 0 ANSWER 0, 4

5. x3–25x2 = 0 ANSWER 0, 5+–

6. c3 – 7c2 + 12c = 0 ANSWER 0, 3, and 4