Factoring Techniques – 2010

1 Factoring is useful

1.1 Prime Factorization vs. Factors

Factoring with numbers allows us to see how we might be able to form them by multiplication. You should be ableto tell the difference between listing the factors of a number of writing it’s prime-factorization.What is the difference between the factors of 24 and the prime factorization of 24?

1.2 Factored Form

Remember factors are just quantities being multiplied.

factor · factor · factor = product

Factors can be simple as in8 · 14 = 112

of they can be complex as in(

2(3) +2

3

)(

2(3)−2

3

)

= 36−4

9= 35

5

9

In order to factor you must be able to tell when an expression is in factored form and when it is in “simplest form”

(x+ 3)(2x+ 4)︸ ︷︷ ︸

= 2x2 + 10x+ 12︸ ︷︷ ︸

Factored Form Simplest Form

For each expression below tell whether it is in factored or simplest form.

1. (x+ 4)(2x− 5)

2. (3x+ 2)− (4x+ 3)

3. x2− 3x+ 10

4. (2x+ 5)(9x− 4) + 3(x+ 2)

5. 4x+ 2

1.3 Solving Equations vs. Factoring

Many exercises that you have completed asked you to “factor” an expression. Here’s an example.Being asked to solve an equation for one variable requires that you isolate that variable and express it’s value. Forexample, in the equation 2x + 5 = 13 the value of x must be 4. So when you are asked to solve 2x + 5 = 13, youanswer simply x = 4. With more complex equations, “solving” still requires this.Here’s an example of an equation that can be solved by factoring

Problem: Factor x2− 10x− 200

Solution:

x2− 10x− 200 = (x− 20)(x+ 10)

︸ ︷︷ ︸

Solution

Figure 1: Example of a ”factor” problem

24x3− 10x2

− 56x = 0 Solve this equation

2x(12x2− 5x− 28) = 0 Factor out the GCF, 2x

2x(12x2− 21x+ 16x− 28) = 0 Split the Middle

2x(3x(4x− 7) + 4(4x− 7)) = 0 Factor by grouping

2x(3x+ 4)(4x− 7) = 0 Factor out (4x− 7) from both terms

x = 0,−4

3,7

4Use the Zero Product Rule, to find solutions

Here are a few more examples

x2− 4x− 5 = 0

(x− 5)(x+ 1) = 0

x = −1, 5

x2 + 6x = −8

x2 + 6x+ 8 = 0

(x+ 4)(x+ 2) = 0

x = −4,−2

x2− 26 = −11x

x2 + 11x− 26 = 0

(x+ 13)(x− 2) = 0

x = −13, 2

These solutions can all be checked by substituting back into the original equation. Here’s an example of the firstequation being checked.

x2− 4x− 5 = 0

x = −1, 5

(−1)2 − 4(−1)− 5 = 0 Check x = −1

1− (−4)− 5 = 0

5− 5 = 0 x = −1 checks

(5)2 − 4(5)− 5 = 0 Check x = 5

25− 20− 5 = 0

0 = 0 x = 5 checks

Solve the following equations, and check your answers by substitution

1. x2− 3x = −2 2. x2

− 4x− 21 = 0 3. 4x+ x2 = 0

2 Factoring Strategies

2.1 Greatest Common Factor (GCF)

Some expressions can be completely factored by simply factoring out the Greatest Common Factor. For example4x+ 8 can be factored to 4(x+ 2), which cannot be further factored. 10x3 + 5x+ 20x2 can be factored by the GCFof the terms 10x3, 5x, and 20x2:

10x3 + 5x+ 20x2 = 5x(2x2 + 1 + 4x)

and that cannot be factored any further. Noticing that the terms of an expression can be divided by a commonfactor is an essential factoring skill.Factor each expression

1. 12x+ 20 2. 3x2 + 18x3 + 23x4 3. 5x+ 10xy + 25x2y2

2.2 Factoring by Grouping

We can factor expressions with four terms by grouping. See how to factor x2 + 4x+ 15x+ 60

x2 + 4x+ 15x+ 60 = (x2 + 4x) + (15x+ 60)

= x(x+ 4) + 15(x+ 4)

= (x+ 4)(x+ 15)

Factor

1. x2 + 5x+ 6x+ 30 2. 12x2 + 39x+ 28x+ 91 3. 20x2 + 5x− 32x− 8

2.3 Sum/Product Rule

2.3.1 Background

Before you learned to factor algebraic expressions you learned to apply the distributive property to multiply poly-nomials. For example:

(x+ 3)(x+ 5) = x2 + 3x+ 5x︸ ︷︷ ︸

+15

= x2 + 8x+ 15

Hopefully you developed a rule to help you multiply simple expressions like these. Simplify each expression

1. (x+ 2)(x+ 4)

2. (x+ 6)(x+ 7)

3. (x+ 4)(x− 4)

4. (x− 3)(x+ 4)

5. (x+ 10)(x− 3)

6. (x+ 3)(x+ 8)

What’s the pattern? Well the simplest form of each expression above looks something like x2 + x+ . The middleterm is the sum of the two numbers, and the last term is the product of the two. In other words, (x + a)(x + b) =x2 + (a+ b)x+ ab. Notice how this is true for the following examples - be careful with the signs (negative/positive)when you add and multiply:

• (x+ 2)(x− 4) = x2− 2x− 8 which is x2 + (2 + (−4))x+ 2(−4)

• (x− 3)(x− 4) = x2− 7x+ 12 which is x2 + (−3 + (−4)) +−3(−4)

• (x+ 10)(x+ 20) = x2 + 30x+ 200 which is x2 + (10 + 20)x+ 10(20)

2.3.2 Factoring x2 + bx+ c with the Sum/Product Rule

An expression like x2− 2x− 8 is in the form x2 + bx+ c where b = −2 and c = −8. When we talk about factoring

the coefficient of the term with x is often referred to as the variable b and the constant term is c. So even if theterms are in different order as in, 5 + x2

− 7x, we could determine that b = −7 and c = 5.If an expression in the form x2 + bx + c we try to use the Sum/Product Rule to factor it. We look for factors of cthat add to b. For example,

Factor: x2− 2x− 8

b = −2 and c = −8, so we are looking for two numbers whose product is −8 and whose sum is −2. The two numbersare −4 and 2.

−4 · 2 = −8 The product is correct

−4 + 2 = −2 The sum is correct

Once we know the two numbers that fit the Sum/Product Rule we can factor the expression

x2− 2x− 8 = (x − 4)(x+ 2)

Sum Product Practice. Find two numbers to fit each case in the table

Write Two Numbers With a Product of And a Sum of:

15 8

36 13

-65 -8

-371 46

Now factor each expression

1. x2 + 10x+ 21

2. x2− 9x− 22

3. x2 + 13x+ 40

4. x2− 20x+ 75

5. x2− 100x+ 1131

6. x2− 2x+ 1

2.4 Factoring expressions of the form ax2+ bx + c, sometimes you have to split the

middle. . .

2.4.1 Sometimes you don’t have to split the middle

When an expression of the form ax2 + bx+ c can be factored using the GCF, you sometimes do not have to split themiddle. Try these:

1. 3x2− 21x− 24 2. 6x2 + 60x+ 144 3. 2x2

− 6x+ 4

2.4.2 Sometimes you have to split the middle

When an expression of the form ax2 + bx+ c cannot be simplified using the GCF, you must try to split the middle.To split the middle follow three steps.

1. Find factors of ac that add to b1

2. Split the middle using these two factors

3. Factor by grouping

Here’s a nice example, factoring 2x2 + 3x+ 1

2x2 + 3x+ 1

2x2 + x+ x+ 1

2x2 + 2x+ 1x+ 1

2x(x+ 1) + 1(x+ 1)

(2x+ 1)(x+ 1)

The middle was split with the two numbers who multiplied to ac or 2(1) = 2 and added to b, or 3. After that step,we factor by grouping.

1This is very similar to the first step of factoring with the Sum/Product Rule

Another example with bigger numbers. Factor 12x2 + 67x+ 91.

12x2 + 67x+ 91

12x2 x+ x+ 91

12x2 + 28x+ 39x+ 91

4x(3x+ 7) + 13(3x+ 7)

(4x+ 13)(3x+ 7)

Your turn. Factor

1. 2x2 + 5x+ 2 2. 15x2 + 31x+ 14 3. 21x2− 37x+ 12