(x + 2)(x + 5) is an area model with four pieces: x², 5x, 2x, and 10, giving x² + 7x + 10.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Factoring is the same picture backwards
To factor x² + 7x + 10, find two numbers multiplying to 10 and adding to 7. Those are 2 and 5.
A pattern worth recognising
x² − 9 is (x + 3)(x − 3). Any difference of two squares factors this way.
Common factors first
Always take out a common factor before anything else. 2x² + 10x is 2x(x + 5).
Two directions of one picture
Multiplying (x + 2)(x + 3) builds x² + 5x + 6; factoring takes it apart again. Both are read off the same area rectangle split into four pieces. Recognising them as one skill in two directions makes factoring much less mysterious.
Every term times every term
(x + 2)(x + 3) needs four products: x², 3x, 2x and 6. The middle terms then combine to 5x. Missing one of the four is the standard expansion error, and drawing the four-box grid prevents it.
Always look for a common factor first
For 2x² + 10x + 12, take out the 2 to get 2(x² + 5x + 6) before factoring further. Pulling out the common factor first keeps the remaining numbers small and is the step most often skipped.
Check factoring by expanding
Multiply your factors back out and compare with the original. This check is complete — if the expansion matches, the factoring is correct — and it is faster than redoing the factoring.
Step 2: Try It Yourself
Tap and try it out.
102
105
10020
5010
- 10 × 10100
- 2 × 1020
- 10 × 550
- 2 × 510
- 12 × 15180
The four pieces cover the whole rectangle, so their areas add to 12 × 15.
Step 3: Watch an Example
One step at a time.
Watch Maya Factor x² + 9x + 20
Maya factors a trinomial.
- Step 1
She needs two numbers multiplying to 20 and adding to 9.
Step 4: Your Turn
Practice makes it stick.
Expanding
Problem 1 of 2
(x + 3)(x + 4) = x² + ?x + 12. What is the middle coefficient?
Factoring
Problem 2 of 2
x² + 8x + 15 factors as (x + 3)(x + ?). What is the missing number?
Both Directions
1 of 4
(x + 2)(x + 6) = x² + ?x + 12. Middle coefficient?
2 of 4
x² + 10x + 21 = (x + 3)(x + ?). Missing number?
3 of 4
x² − 25 = (x + 5)(x − ?). Missing number?
4 of 4
3x² + 12x = 3x(x + ?). Missing number?
Step 5: Quick Check
Show what you know.
Question 1 of 2
x² + 11x + 24 = (x + 3)(x + ?). Missing number?
Question 2 of 2
What should you always look for first when factoring?
What You Learned
- Multiplying binomials is an area model with four pieces.
- Factoring runs the same picture backwards.
- Take out a common factor first, and watch for a difference of squares.