Same Math › Algebra 1 · grades 8–10 › Adding, subtracting and multiplying polynomials
How do kids add and multiply polynomials?
Adding polynomials means combining like terms: x² with x², x with x, numbers with numbers. Subtracting is the same after the minus sign flips every sign in the second polynomial. Multiplying means every term in one times every term in the other. Many classes use a box (area model): each term gets a row or column, fill in the products, then combine like terms.
- Corners: the first and last products.
- Middle boxes: like terms, combined into one.
- Every box filled means nothing was missed.
Why schools teach it
The algebra behind factoring
Algebra 1 students add, subtract and multiply polynomials and see that the answer is always another polynomial (HSA.APR.A.1). It's the same box that comes back later for factoring, run in reverse.
FOIL (First, Outer, Inner, Last) works for two binomials only. The box keeps working for bigger problems, which is why many teachers start there.
At the kitchen table
What to say, and what to skip
Say this
Which terms are like terms?
Same letter, same exponent.When you subtract, what happens to each sign in the second one?
They all flip.How many boxes do you need?
Terms in the first times terms in the second.Which boxes are like terms?
Usually the diagonal.
Skip this
- Flipping only the first sign when subtracting.−(3x² − 2x + 5) is −3x² + 2x − 5.
- Adding exponents when adding.x² + x² = 2x², not x⁴. Exponents add only when you multiply.
- (x + 3)² = x² + 9.Draw the box: (x + 3)(x + 3) has a 6x in the middle.
See it in action
Multiplying (x + 4)(x − 7)
Fill the box
x², −7x, 4x, −28Combine the middle
−7x + 4x = −3xWrite it in order
x² − 3x − 28
Check with x = 1: (5)(−6) = −30 and 1 − 3 − 28 = −30 ✓.
Free practice sheet
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Parents also ask
What's standard form for a polynomial?
Highest exponent first: 2x² + x − 15.
Is FOIL wrong?
No, it's the box in a different order. It only works for two binomials, though.
What about a binomial times a trinomial?
A 2-by-3 box: six products, then combine like terms.
How can you check?
Plug in a small number, like x = 1, into the problem and the answer. They should match.