**Warm-up**

1. Find the area of a square whose sides are 12 m long.

2. Find the product of 2x(x + 1).

3. Find the product (2x + 1) (x + 1).

4. If 2ab = 0 and a = -1, what is the value of b?

5. What is the greatest common factor of 6, 15, and 21?

**Answers to warm-up
**1. Area is 144 m

2. 2x

3. 2x

4. b = 0

5. GCF is 3

A trinomial is an expression that can be written as the sum of 3 unlike terms

Factoring is undoing FOILing. The trinomial will factor
into two binomials.

Examples 1-4 show how to factor when a = 1. Example 5 shows how to factor when a
≠1.

Factoring when a = 1

• If c is positive, then the factors are either both
positive or both negative.

o If b is positive, then the factors are positive

o If b is negative, then the factors are negative.

o In either case, you're looking for factors that add to b.

• If c is negative, then the factors you're looking for are of alternating
signs; that is, one

is negative and one is positive.

o If b is positive, then the larger factor is positive.

o If b is negative, then the larger factor is negative.

In either case, you're looking for factors that are b units apart.

**Example 1**

Factor x^{2} - 9x + 18 (ax^{2} + bx + c)

Solution | |

(x ____)(x ____) | |

List the factors of 18 | (the c) |

-9 ·- 2 | Factors must add up to –9. So, one factor + and one factor is - |

2 · 9 | |

-6 · -3 | |

3 · 6 | |

-18 · -1 | |

1 · 18 |

Guess and check

( x-9)( x-2) =x^{2}-11x+18 ←no

( x+9)( x-2) =x^{2}+11x+18 ←no

(x- 3)( x-6) =x^{2}-3x-6x+18←yes

Example 2

Factor x^{2} +3x + 2

Solution | |

(x ____)(x ____) | |

Factors of 2 | Add up to 3 |

2·1 | Yes |

1·2 | Yes |

Guess and check

(x + 2)(x +1) = x^{2} + 2x + x +1= x^{2} +
3x +1
←
yes!

**Example 3**

Factor x^{2} -3x - 40

Solution | |

(x ____)(x ____) | |

Factors of –40 | Add up to -3 |

-8 · 5 | Yes |

-5 · -8 | No |

-2· 20 | No |

-20 · 2 | No |

(x-8)(x+5)

Check

(x-8)(x+5)=x^{2} -8x +5x-40= x^{2} -3x - 40

**Example 4
**

Factor x

Solution | |

(x ____)(x ____) | |

Factors of –6 | Add up to -1 |

-2 · 3 | No |

-3 · 2 | Yes |

(x-3)(x+2)

Check

(x-3)(x+2)=x^{2}-3x +2x- 6= x^{2}-x - 6

Sometimes a (the x^{2}coefficient) is not 1.

**Example 5
**

Factor 2x

Solution

x

Use guess and check!

Factoring when a ≠1

**Example 1**

Factor 3x^{2} +11x +10

Solution

Factors of | Factors of c | Product |

3 • 1 | 10 • 1 | (3x + 10)(x + 1) |

1 • 3 | 10 • 1 | (x + 10)(3x + 1) |

3 • 1 | 5 • 2 | (3x + 5)(x + 2) →
3x^{2} + 11x + 10 ←This is correct! |

1 • 3 | 5 • 2 | (x + 5)(3x + 2) |

(3x + 5)(x + 2)

**Example 2
**

Factor 4x

Solution

Factor out the greatest common factor.

2(2x

Factors of | Factors of c | Product |

2 • 1 | -7 • 1 | 2(2x – 7)(x + 1) →
4x^{2}– 10x – 14 ←This is correct! |

1 • 2 | -7 • 1 | 2(x – 14)(4x – 1) |

2 • 1 | 7 • -1 | 2(2x – 14)(2x + 1) |

1 • 2 | 7 • -1 |

(4x – 14)(x + 1)

**Special Factoring Patterns – Yellow box on page 208**

To factor a difference of two squares: | Examples |

a^{2} −b^{2} = (a + b)(a − b) |
9x^{2}−100 = (3x +10)(3x −10) |

To factor a perfect square trinomial: | |

a^{2} + 2ab + b^{2} = (a + b)^{2} |
16x^{2} + 20x +169 = (4x +13)^{2} |

a^{2} − 2ab + b^{2} = (a − b)^{2} |
4x^{2} − 20x + 25 = (2x − 5)^{2} |

**Guidelines for factoring completely – Yellow box on
page 210**

1. Factor out the greatest common factor first.

2. Look for a difference of two squares.

3. Look for a perfect square trinomial.

4. If a trinomial is not a perfect square, use trial and error to look for a

pair of factors.

**Example 3**

Factor 25x^{2} - 81

Solution

Test whether the expression is a difference of two squares. Ask these questions:

• Is the expression a difference? Yes

• Is the first term a square? Yes

• Is the second term a square? Yes

To factor a difference of two squares:

a^{2} −b^{2} = (a + b)(a − b)

(5x + 9)(5x – 9)

**Example 4**

Factor 16x^{2} + 56x + 49

Solution

Test whether the trinomial is a perfect square trinomial. Ask these questions:

• Is the first term a square? Yes

• Is the last term a square? Yes

• Is the middle term twice the product of
? Yes, 4 • 7 = 28, which is ½ of 56.

To factor a perfect square trinomial:

a^{2} + 2ab + b^{2} = (a + b)2

a^{2} − 2ab + b^{2} = (a − b)2

(4x + 7)^{2}

Solving equations by factoring

Some quadratic equations can be solved by factoring.

• First the equation must be written in standard form, a x^{2} + bx + c

• Then, if the trinomial is factorable, the equation can be solved using
the zero-product

property.

Zero-product property (ZPP)

If ab = 0, then a = 0 or b = 0 or both are zero.

Example: If y (x + 5) = 0, then y = 0 or x + 5 = 0, or both.

**Example 6
**

Solve 6x = x

Solution

x

The equation is a perfect square trinomial.

To factor a perfect square trinomial:

a

a

So, the equation factors into

(x + 3)(x + 3) = 0

Set each factor equal to zero (ZPP) and solve for x.

x + 3 = 0 or x + 3 = 0

x = -3 or x = -3

**Example 7**

Solve 8x^{2} - 18x = -4

Solution

8x^{2} - 18x + 4 = 0 ←Rewrite equation in
standard form.

2(4x^{2} - 9x + 2) = 0 ←Factor out the
greatest common factor, 2.

4x^{2} - 9x + 2 = 0 ←Divide both sides by 2

(4x – 1)(x – 2) = 0 ←Factor the trinomial

4x – 1 = 0 or x – 2 = 0 ←Set each factor equal to 0
and solve for x. (ZPP)

x = ¼ or x = 2

There where three factors that multiplied to 0. Why didn’t we set all three of
them equal to

zero when solving?