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
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