Factoring Intuition Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Factoring Intuition.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Understanding factoring as finding what multiplies together to give an expression.

Reverse engineering multiplication: 'What times what gives x2+5x+6x^2 + 5x + 6?'

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Factoring intuition asks 'what times what produces this?' so a sum reappears as a product.

Common stuck point: The procedure for factoring intuition is the easy part; the trap is finding numbers that add to b but ignore the product. Asking "Am I turning a sum of terms into a product of simpler factors?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I turning a sum of terms into a product of simpler factors?

Worked Examples

Example 1

easy
What two numbers multiply to 12 and add to 7?

Answer

3 and 43 \text{ and } 4

First step

1
List factor pairs of 12: (1,12),(2,6),(3,4)(1,12), (2,6), (3,4).

Full solution

  1. 2
    Check sums: 1+12=131+12=13, 2+6=82+6=8, 3+4=73+4=7 ✓
  2. 3
    The numbers are 3 and 4.
Factoring intuition is about finding what multiplies together to give an expression. For trinomials x2+bx+cx^2 + bx + c, you need two numbers with product cc and sum bb.

Example 2

medium
Find two numbers that multiply to −15-15 and add to 22.

Example 3

easy
Factor 3x2+12x3x^2+12x.

Example 4

medium
Factor x2+x−20x^2+x-20.

Example 5

medium
Factor x2−36x^2-36.

Example 6

medium
Factor x2+6x+9x^2+6x+9.

Example 7

medium
Factor 4x2−94x^2-9.

Example 8

hard
Factor 2x2+7x+32x^2+7x+3.

Example 9

hard
Factor x4−16x^4-16 completely.

Example 10

hard
Factor by grouping: x3+2x2+3x+6x^3+2x^2+3x+6.

Example 11

challenge
Factor x4+x2−12x^4+x^2-12 completely over the integers.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Find two numbers that multiply to 18 and add to 9.

Example 2

hard
Find two numbers that multiply to −24-24 and add to −2-2.

Example 3

easy
Factor x2+5x+6x^2+5x+6.

Example 4

easy
Factor out the greatest common factor of 6x+96x+9.

Example 5

easy
Factor x2−16x^2-16.

Example 6

easy
Factor x2+7x+12x^2+7x+12.

Example 7

easy
Factor x2−5x+6x^2-5x+6.

Example 8

easy
Factor 2x2+4x2x^2+4x.

Example 9

easy
Is factoring the same as solving an equation?

Example 10

easy
Factor x2−1x^2-1.

Example 11

medium
Factor 2x2+7x+32x^2+7x+3.

Example 12

medium
Factor x3−xx^3-x completely.

Example 13

medium
Factor 4x2−12x+94x^2-12x+9.

Example 14

medium
Factor x2−x−12x^2-x-12.

Example 15

medium
Factor 3x2−273x^2-27.

Example 16

medium
Factor by grouping: x3+2x2+3x+6x^3+2x^2+3x+6.

Example 17

medium
Why does x2+1x^2+1 not factor over the real numbers?

Example 18

challenge
Factor x4−16x^4-16 completely over the reals.

Example 19

challenge
Factor x2+(a+b)x+abx^2+(a+b)x+ab symbolically.

Example 20

challenge
Use factoring to simplify x2−x−6x2−9\frac{x^2-x-6}{x^2-9} (state restrictions).

Example 21

medium
Factor x2+2x−8x^2+2x-8.

Example 22

medium
Factor 9x2−259x^2-25.

Example 23

easy
Find two numbers that multiply to 2020 and add to 99.

Example 24

easy
Factor x2+8x+15x^2+8x+15.

Example 25

easy
Factor x2−25x^2-25.

Example 26

easy
Factor x2−9x+20x^2-9x+20.

Example 27

easy
Factor x2+2x−15x^2+2x-15.

Example 28

medium
Factor 2x2+10x+122x^2+10x+12 completely.

Example 29

medium
Factor x2−3x−28x^2-3x-28.

Example 30

medium
Factor x2−10x+25x^2-10x+25.

Example 31

medium
Factor x2+5x−14x^2+5x-14.

Example 32

hard
Factor 3x2−10x−83x^2-10x-8.

Example 33

hard
Factor 6x2+11x−106x^2+11x-10.

Example 34

hard
Factor x2+10xy+25y2x^2+10xy+25y^2.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

multiplicationdistributive property