➕ Algebra 1 · Exponents and Polynomials

Multiplying Polynomials: notes and common mistakes

Study notes

To multiply polynomials, multiply every term in one factor by every term in the other, then combine like terms. A factor with m terms times a factor with n terms gives m × n products before combining.
Monomial times monomial: multiply the coefficients, and add the exponents of each variable that appears in both.
FOIL is a memory aid for exactly two binomials. It is not a separate rule.
For two nonzero polynomials in one variable, the degree of the product is the sum of the degrees. The product of two polynomials is always a polynomial.
Check an expansion with a test value such as x = 3 in the original and in the answer. Avoid 0, 1 and 2.

Write each factor’s terms with their signs.
Multiply every term in the first factor by every term in the second (distribute, or use a box). Count the products.
Add the products and combine the like terms.
Write the answer in standard form, highest power first.
Check the degree (it should be the sum of the degrees), then check with a test value such as x = 3.

Monomial times monomial: (ax^m)(bxⁿ) = (a × b)x^m+n
Monomial times polynomial: a(b + c + d) = ab + ac + ad
Two binomials: (a + b)(c + d) = ac + ad + bc + bd
Binomial times trinomial: (a + b)(c + d + e) = ac + ad + ae + bc + bd + be
Degree of a product: degree of the product = degree of the first factor + degree of the second (both nonzero, one variable)

Coefficients multiply and exponents add: 5x³ × 2x⁴ = 10x⁷, not 7x⁷ and not 10x¹². Exponents add only for the same variable.
Multiply every pair. (x + 7)(x + 4) is not x² + 28, because the Outer and Inner products are missing.
Signs travel with their terms. In (x − 4) the second term is −4.
FOIL fits only two binomials. A binomial times a trinomial has 6 products.
A power of a binomial is repeated multiplication: (x + 3)² = x² + 6x + 9, not x² + 9.
Only like terms combine: x² and x stay separate.

Why every term times every term? Distribute twice: (a + b)(c + d) = a(c + d) + b(c + d) = ac + ad + bc + bd. Each term of the first factor ends up multiplied by each term of the second.
Why add exponents? x² × x³ is 2 factors of x times 3 more, which is 5 factors: x⁵.
Why is the product always a polynomial? Multiplying monomials only adds whole-number exponents, so the exponents stay whole numbers. Polynomials are closed under multiplication, just as the integers are.

✗ Mistake(x + 3)(x + 5) = x² + 15 (multiplying only the first terms and the last terms).

✓ InsteadEvery term meets every term. The Outer and Inner products matter too: x² + 5x + 3x + 15 = x² + 8x + 15.

✗ Mistake5x³ × 2x⁴ = 10x¹² (multiplying the exponents) or 7x⁷ (adding the coefficients).

✓ InsteadMultiply the coefficients and add the exponents: 5x³ × 2x⁴ = 10x⁷.

✗ Mistake(x + 3)² = x² + 9 (the exponent goes onto each term).

✓ Instead(x + 3)² means (x + 3)(x + 3), which is x² + 6x + 9. An exponent does not distribute over a sum.

✗ MistakeFOIL works for every product of polynomials.

✓ InsteadFOIL lists the four products of two binomials only. For a binomial times a trinomial (6 products), use every term times every term.

✗ MistakeThe degree of a product is the product of the degrees.

✓ InsteadThe degrees add. A degree-2 polynomial times a degree-3 polynomial has degree 5, not 6.

✗ Mistakex × x³ = x³ (a lone x has no exponent, so it adds nothing).

✓ InsteadA lone x has exponent 1, so x × x³ = x¹⁺³ = x⁴.

Explain to a friend (or a pet!) how to multiply (x − 5)(3x + 2), why there are four products, and how you would check your answer.

Explaining it out loud shows what you really know. Did you…

I multiplied every term by every term: 3x², 2x, −15x and −10.
I combined the like terms, 2x − 15x = −13x, and wrote 3x² − 13x − 10 in standard form.
I checked with x = 4: (4 − 5)(12 + 2) = −14, and 48 − 52 − 10 = −14.

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