Same Math › Algebra 1 · grades 8–10 › Factoring with the box
How does the box method factor?
Factoring with the box is multiplying with the box, run backwards. To factor x² + 7x + 12, put x² in the top-left box and 12 in the bottom-right. Find two terms for the other boxes that add to 7x and multiply like the corners (3x and 4x), then read the factors off the edges: (x + 3)(x + 4).
- Corners first: the x² term top-left, the number bottom-right.
- The other two boxes split the middle term and must add to it.
- The edges are the factors. Each box is its row edge times its column edge.
Why schools teach it
It turns guessing into a puzzle with a check built in
Algebra 1 students rewrite quadratics in factored form to find where they equal zero (HSA.SSE.A.2 and B.3a). Many of us learned to factor by trial and error: guess two binomials, FOIL them, and try again.
The box gives every piece a place. Kids who multiplied binomials with the box in seventh or eighth grade already know the layout, so factoring becomes filling in a box whose total is known. Every box can be checked by multiplying its edges.
At the kitchen table
What to say, and what to skip
Say this
What's the greatest common factor? Pull it out first.
Then factor what's left.Which two boxes are already known?
The first and last terms.What two terms add to the middle term?
Does each box equal its edges multiplied?
That's the check.
Skip this
- Teaching FOIL backwards or guess-and-check.The box is the same work, organized.
- Forgetting the GCF.6x² + 12x + 6 starts with 6(x² + 2x + 1).
- Dropping a negative sign from an edge.If the bottom-right is −12, one edge is negative.
See it in action
Factoring x² − 2x − 15
Put x² top-left and −15 bottom-right
Find two terms that multiply to −15x² and add to −2x
−5x and 3xPut them in the empty boxes and find the edges
x and −5 across, x and 3 downWrite the factors
(x − 5)(x + 3)
Check: multiply the edges back. x² + 3x − 5x − 15 = x² − 2x − 15.
Free practice sheet
Make a factoring box sheet
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Parents also ask
How do you find the edges once the boxes are full?
Take the greatest common factor of each row and each column. The top-left box x² gives x and x; the rest follow.
Does the box work when the x² term has a number, like 3x²?
Yes. Put 3x² in the corner and find two middle terms that multiply to 3 × (the last number) times x². The edges come out as (3x + …) and (x + …).
What if no two numbers work?
Then the expression doesn't factor with whole numbers. Classes say it's prime, and later use the quadratic formula.
Is the box the same as the X method?
They find the same two numbers. The X finds them; the box shows where they go. Many classes use both together.