Lesson · Polynomials and Factoring · Polynomial arithmetic

Multiplying Polynomials

One rule covers every polynomial product: distribute until no brackets remain. FOIL is that rule applied to two binomials and is not worth memorising as a separate fact.

Open the interactive lesson Practice

xᵐ · xⁿ = xᵐ⁺ⁿ exponents ADD when multiplying (a + b)(c + d) = ac + ad + bc + bd m terms × n terms → mn products, before collecting

Distribution, and the count it produces

Core ideas

What this lesson establishes

Multiplication looks harder than addition and involves one more idea. It is also more mechanical: there is nothing to decide.

Coefficients multiply, exponents add

(3x²)(4x⁵) = 12x⁷. The coefficients are just numbers, so they multiply. The powers add because x²·x⁵ is two xs times five xs — seven xs. That is Foundations of Algebra's exponent law, unchanged.

The distributive law is the entire method

Multiply each term of the first polynomial by each term of the second, then collect like terms. Every layout — FOIL, the grid, the vertical arrangement — is bookkeeping for that one instruction.

The product's degree is the sum of the degrees

Degree 3 times degree 4 gives degree 7, always. The leading terms multiply to give the leading term, and two nonzero numbers cannot multiply to zero, so it can never cancel away.

The material

Multiplying, from one term to many

Monomials first, then the general rule, then how to keep track of the pieces.

A single term times a single term

Multiply the coefficients; add the exponents of each variable separately.

(3x²)(4x⁵) = 12x⁷ (−2x³y)(5xy⁴) = −10x⁴y⁵

In the second line the x exponents give 3 + 1 = 4 and the y exponents give 1 + 4 = 5. Each variable is tracked on its own; they never mix.

A term times a polynomial

This is the distributive law directly: a(b + c) = ab + ac, with as many terms inside the bracket as you like.

3x(2x² − 5x + 4) = 6x³ − 15x² + 12x

The 3x reaches every term inside the bracket — the same "reaches every term" discipline as the minus sign in “Adding and Subtracting Polynomials”.

A polynomial times a polynomial

Theorem · The general product

To multiply two polynomials, multiply each term of the first by each term of the second and add all the results.

A polynomial with m terms times one with n terms produces mn products before any collecting. Counting them first tells you when you have finished.

For two binomials that count is 2 × 2 = 4, and the four products are traditionally remembered as FOIL — First, Outer, Inner, Last:

(x + 4)(x − 7) First x · x = x² Outer x ·(−7) = −7x Inner 4 · x = 4x Last 4 ·(−7) = −28 collect = x² − 3x − 28

FOIL is worth knowing as a memory aid and worth distrusting as a method, because it describes only the 2 × 2 case. Asked for (2x − 3)(x² + 4x − 5) there is no "outer" and "inner" to speak of — there are 2 × 3 = 6 products. The counting rule survives where the acronym does not.

Keeping track

Take the terms of the first polynomial one at a time and distribute each across the whole second polynomial. Write each partial product on its own line, then collect down the columns. Six products written on two lines are far easier to check than six products written in a row.

Example · Degree, predicted

(2x − 3)(x² + 4x − 5) has degree 1 + 2 = 3, and its leading term is 2x · x² = 2x³ before you multiply anything else. If your answer is not a cubic starting 2x³, something went wrong — and you know it without checking the rest.

Interactive

Multiplying

Every term against every term, then collected

Method

Multiplying two polynomials

Predict the shape of the answer first; it makes the check free.

Predict the degree and the leading term

Add the degrees; multiply the leading coefficients. You now know the first term of the answer before doing any work, and you will know immediately if the result is malformed.

Count the products you owe

m terms times n terms is mn products. Write the number down. Missing one term is the second most common error here, and a count catches it.

Distribute one term at a time, on its own line

Take the first term of the first polynomial across the whole of the second, then the next. Never work two terms at once in your head.

Collect like terms and put the answer in standard form

Collect by power, highest first, exactly as in “Adding and Subtracting Polynomials”. Then check the leading term against your prediction from step 1.

Products

Score 0 / 0

Choose the answer you can justify.

A binomial times a trinomial

(2x − 3)(x² + 4x − 5). Degree 1 + 2 = 3; leading term 2x³; 2 × 3 = 6 products owed.

2x across the trinomial: 2x · x² = 2x³ 2x · 4x = 8x² 2x ·(−5) = −10x −3 across the trinomial: −3 · x² = −3x² −3 · 4x = −12x −3 ·(−5) = +15 collect: 2x³ + (8 − 3)x² + (−10 − 12)x + 15 answer 2x³ + 5x² − 22x + 15

Six products written, six products owed. The leading term is 2x³ as predicted, and the constant term is (−3)(−5) = 15 — which is always just the product of the two constant terms, another free check.

At x = 1: the factors give (2 − 3)(1 + 4 − 5) = (−1)(0) = 0, and the answer gives 2 + 5 − 22 + 15 = 0. Agreement.

For a faded product, take (x + 2)(2x² − x + 3). The supplied prediction is degree 3, leading term 2x³, and 2 × 3 = 6 partial products. Write those six products, collect, and test x = 1. The result is 2x³ + 3x² + x + 6, and both forms give 12; a missing x² term means one of the two middle products was skipped.

Common mistakes

Multiplying the exponents instead of adding themx²·x⁵ = x⁷, not x¹⁰. Multiplying exponents is the rule for a power of a power, (x²)⁵ = x¹⁰, and the two get swapped under pressure.

Trusting FOIL beyond two binomialsFOIL names the four products of a 2 × 2 case. For anything larger it quietly suggests you are finished when you are not. Count mn instead.

Forgetting to collect after distributing(x + 4)(x − 7) is not x² − 7x + 4x − 28 as a final answer. The middle terms are like terms and must be combined: x² − 3x − 28.

Check your understanding

✓

Finish when you can state the degree and leading term before you multiply.

For any product you should be able to say the degree, the leading term, the constant term and the number of partial products before writing a line of working — and then use all four to check the result. That is what makes “Factoring Out the Greatest Common Factor” onward, where you run this process backwards, feel like a search rather than a guess.

Note: Everything from here to “Factoring Special Forms” is this lesson in reverse. Factoring is asking which two polynomials would multiply to give the one in front of you, so the fluency you build expanding is exactly the fluency that makes factoring checkable.