Factoring trinomials

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Intros
Lessons
  1. Use "cross-multiply, then check" method to factor a trinomial
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Examples
Lessons
  1. Factor:
    1. b2−b−20{b^2} - b - 20
    2. x2−10x+16{x^2} - 10x + 16
    3. 2x3−14x2+24x2{x^3} - 14{x^2} + 24x
    4. 14+5y−y214 + 5y - {y^2}
  2. Factor:
    1. 2x2+25x+122{x^2} + 25x + 12
    2. 5x2+8x+35{x^2} + 8x + 3
    3. 8x2+10x−38{x^2} + 10x - 3
    4. 6m2−13m−86{m^2} - 13m - 8
    5. 18x2−9x+118{x^2} - 9x + 1
    6. 63+20z−3z263 + 20z - 3{z^2}
    7. 8x2+8x−68{x^2} + 8x - 6
  3. Factor:
    1. 8x2+xy−9y28{x^2} + xy - 9{y^2}
    2. 6x2+17xy−3y26{x^2} + 17xy - 3{y^2}
    3. 14x2−4xy+16y2\frac{1}{4}{x^2} - 4xy + 16{y^2}
  4. Factor:
    1. 8x2y2−xy−98{x^2}{y^2} - xy - 9
    2. 25sin⁡2x−35sin⁡x+1225{\sin ^2}x - 35\sin x + 12
    3. 12cos⁡2x−20cos⁡x+312{\cos ^2}x - 20\cos x + 3
    4. 6(x+5)2+17(x+5)−36{\left( {x + 5} \right)^2} + 17\left( {x + 5} \right) - 3
    5. 15(x−3)2−11(x−3)−1415{\left( {x - 3} \right)^2} - 11\left( {x - 3} \right) - 14
  5. Factor:
    1. x6−17x3+30{x^6} - 17{x^3} + 30
    2. a4−2a2−63{a^4} - 2{a^2} - 63
    3. 15x4−16x2−1515{x^4} - 16{x^2} - 15
    4. 8x4−14x2+38{x^4} - 14{x^2} + 3
Topic Notes
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How to factor trinomials

Trinomials are algebraic expressions that has three terms in it. Quadratic trinomials are in the form of ax2{x^2} + bx + c, and the a, b, and c all stands for a number.

In order to factor trinomials, you'll have to work to find two numbers that will multiply to equal the "c" from the quadratic form above, and also add up to equal "b". These are the steps for the easier questions where the first "a" is equalled to 1. For harder problems, "a" will be a number that is not one. You'll have to first multiply "a" and "c", and then find factors of the product "a*c" that also add up to "b".

We'll explore this with example questions to demonstrate how to factor trinomials.

 

Question 1

Factor: b2−b−20{b^2} - b - 20

Solution:

In this example, we are going to use the "cross-multiply, then check" method to factor the trinomial. This is one of the ways to factor trinomials.

Use cross multiply and check to factor trinomial
Use cross multiply and check to factor trinomial

Factor the first term b2{b^2}, put the results (b and b)  in the first box.

Factor the term b^2 and put them into the first box
Factor the term b^2 and put them into the first box

Factor the last term -20. There are a few combinations (1x20, 2x10, 4x5) that can give us 20, so which one is it? We'll cross-multiply those combinations to the factors of the first term. See which combination will produce an answer that matches the middle term (in this question, the middle term is –b).

See which combination will produce the answer to match the middle term
Combination that matches the middle term

Out of those combinations, 4x-5 (which equals -1) is able to produce the matching middle term -b.

Find out that 4,-5 match the middle term
Find out that 4,-5 match the middle term

After all the steps, we successfully factored the trinomial  b2−b−20{b^2} - b - 20 into (b+4)∙(b−5)\left( {b + 4} \right) \bullet \left( {b - 5} \right)

Successfully factored b^2-b-20
Successfully factored b^2-b-20

Question 2:

Factor: 2x2+25x+122{x^2} + 25x + 12

Solution:

The "cross-multiply, then check" method can also be used on harder trinomials in which the leading coefficient is not 1. In this question, the leading coefficient is 2 (from the leading term 2x22{x^2}).

Now, we factor the first term 2x22{x^2}. The result is 2x and x. we put them in the first box

Factor the first term 2x^2 and put into the first box
Factor the first term 2x^2 and put into the first box

In order to factor the last term +12, there are a few combinations (1x12, 2x6, 3x4). We'll once again cross-multiply those combinations to the factors of the first term. See which combination will produce a matching middle term (in this question, the middle term is +25x)

Find the combination that matches the middle term 25x
Find the combination that matches 25x

Out of those combinations, 1x12 is able to produce the matching middle term +25x. That is because (2x x 12) + (x x 1) = 24x + x = 25x

Find out that 12 and 1 match the middle term of 25x
Find out that 12 and 1 match the middle term of 25x

After all the steps, we successfully factored the trinomial  2x2+25x+122{x^2} + 25x + 12 into (2x+1)∙(x+12)\left( {2x + 1} \right) \bullet \left( {x + 12} \right)

Factored 2x^2+25x+12 successfully
Factored 2x^2+25x+12 successfully

Factoring perfect square trinomials

As you do more questions that require you to do trinomial factoring, you may run into perfect square trinomials. These are trinomials that has both the "a" and "c" terms being perfect squares and the middle "b" being twice the product of the first and last term. Therefore, after factoring, you'll get an answer in the form of either (a+b)^2 or (a-b)^2.

 

There are good examples for you to further practice factoring trinomials. We've found some excellent online practice questions for you to try that keeps the answers hidden until you hover over the word "answer". There is also an online trinomial calculator for when you have a quadratic equation. It can help you factor any trinomial instantly! Use it to help you check your work.

Basic Concepts
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