Multiplying Polynomials

To multiply polynomials, you use one rule over and over: every term in the first factor multiplies every term in the second factor. Then you combine like terms. A monomial times a monomial, a binomial times a binomial, even a binomial times a trinomial: the sizes change, but the rule does not. “FOIL” is a memory aid for just one case, two binomials.

Polynomial words you will use

A term is one piece of an expression, separated from the others by + or −, and it keeps the sign in front of it, like 5, x, −3x² or 4xy. A monomial is a single term whose variables have whole-number exponents. So x² is a monomial, but 1/x is not. A polynomial is a monomial or a sum of monomials.

• A binomial has two terms, like x + 3. A trinomial has three terms, like x² − 3x + 5.
• The degree of a term is the sum of the exponents of its variables. So −3x² has degree 2, 4xy has degree 2, and a nonzero constant like 7 has degree 0. The degree of a polynomial is the highest degree among its terms, after like terms are combined.
• Standard form (for one variable) lists the terms from the highest degree to the lowest, like x³ − x² − x + 10.

Monomial times monomial

Multiply the coefficients. Then, for each variable that appears in both monomials, add its exponents (the product rule from Laws of Exponents). A variable that appears in only one monomial is simply copied. For example:

(3x²)(−4x⁵) = 3 × (−4) × x² × x⁵ = −12x⁷.

You may regroup like this because multiplication can be done in any order. Three details:

• A variable with no written exponent has exponent 1, so x × x³ = x¹ × x³ = x⁴.
• Exponents add only for the same variable. Different variables just sit side by side: (2x)(5y) = 10xy, and (2x²y)(3xy⁴) = 6x³y⁵.
• Coefficients multiply and exponents add. Do not add the coefficients, and do not multiply the exponents.

Monomial times polynomial: distribute

The distributive property works for a sum with any number of terms: a(b + c + d) = ab + ac + ad. Multiply the monomial by each term, one at a time, using the monomial rule for each product.

2x(x² − 5x + 4) = 2x × x² + 2x × (−5x) + 2x × 4 = 2x³ − 10x² + 8x.

Each term keeps its sign: the middle term is −5x, so its product is −10x². Here nothing combines afterward, because every product has a different power of x.

Binomial times binomial: distribute twice

To multiply (x + 3)(x + 5), distribute the first binomial across the second. This is the rule (b + c)a = ba + ca, with a standing for the whole group (x + 5):

(x + 3)(x + 5) = x(x + 5) + 3(x + 5).

Now distribute again inside each piece: x² + 5x + 3x + 15. Combine like terms: x² + 8x + 15.

Each of the 2 terms in the first factor met each of the 2 terms in the second, so there are 2 × 2 = 4 products. Signs travel with their terms: in (2x − 3)(x + 4), the first factor’s terms are 2x and −3.

The box (area) model

A box keeps the products organized. Write the terms of one factor across the top and the terms of the other down the side, each with its sign. Fill each cell with the product of its row and column. Then add all the cells and combine like terms.

For (x + 3)(x + 5) the cells are x², 5x, 3x and 15. As a picture, it is a rectangle with sides x + 3 and x + 5 cut into four pieces, so the whole area is the sum of the pieces. The rectangle needs x to be a positive length, but the box method works for any numbers, including negative terms: multiply the signs in each cell.

If both factors are in standard form with no missing powers, like terms line up along the diagonals that run from lower left to upper right. With missing powers they may not line up, so scan every cell.

FOIL: a memory aid for two binomials

FOIL stands for First, Outer, Inner, Last. For two binomials, multiply the first terms, the outer terms, the inner terms and the last terms, then add. For (x + 3)(x + 5): First x × x = x², Outer x × 5 = 5x, Inner 3 × x = 3x, Last 3 × 5 = 15.

These are exactly the four products from distributing twice. FOIL is a memory aid, not a new rule. It describes only two binomials. It does not describe a binomial times a trinomial (which has 6 products) or a monomial times a polynomial. For those, use the rule underneath it: every term times every term.

Bigger products

The rule does not change when the factors get bigger. If one factor has m terms and the other has n terms (each factor already simplified), you write m × n products before combining like terms. A binomial times a trinomial gives 2 × 3 = 6 products, and a trinomial times a trinomial gives 3 × 3 = 9.

Count your products. If you wrote fewer than m × n, one is missing. (After combining like terms, the answer can have fewer terms than the number of products.) A box with 2 rows and 3 columns has 6 cells, which makes it a good way to keep track.

Collect like terms and use standard form

When all the products are written, combine like terms by adding their coefficients. Like terms have the same variable part: the same letters with the same exponents. Then write the answer in standard form, highest power first.

Only like terms combine. In x² + 8x + 15 the terms have different variable parts, so the answer stays three terms. Sometimes more combines than you expect: (x + 6)(x − 6) = x² − 6x + 6x − 36 = x² − 36, where the two middle terms cancel. The next lesson, Special Products of Binomials, looks at products like this one.

The degree of a product

For two nonzero polynomials in one variable, the degree of the product is the sum of the degrees of the factors. A degree-2 polynomial times a degree-3 polynomial has degree 2 + 3 = 5, not 6.

Why? The highest-power term in the product comes from multiplying the highest-power terms of the two factors, like x² × x³ = x⁵. No other pair of terms makes x⁵, so nothing can cancel it, and its coefficient is a product of two nonzero numbers, so it is not zero.

The degree gives a quick check. (x + 2)(x² − 3x + 5) should have degree 1 + 2 = 3, and the answer x³ − x² − x + 10 does. If your answer has a different degree, a product went wrong.

One more fact: the product of two polynomials is always another polynomial, because multiplying monomials only adds whole-number exponents. Polynomials are closed under multiplication, just as multiplying two integers always gives an integer.

A power of a binomial is repeated multiplication

An exponent on a binomial means the whole binomial is repeated as a factor: (x + 3)² = (x + 3)(x + 3). Multiply as usual: x² + 3x + 3x + 9 = x² + 6x + 9.

It is not x² + 9, because an exponent does not distribute over a sum. Try x = 3: (3 + 3)² = 36, but 3² + 9 = 18. The next lesson shows a pattern that makes squaring a binomial fast.

Check with a test value

Replace the variable with a number in the original product and in your answer. The two values must match. Pick a number that does not hide mistakes:

• At x = 0, every term that contains x becomes 0, so a missing x-term goes unnoticed.
• At x = 1, every power of x equals 1, so a wrong exponent goes unnoticed.
• At x = 2, x² and 2x are both 4, so the common mix-up of squaring with doubling goes unnoticed.

Use x = 3 (or 4 or 5). No single value catches every mistake, but these avoid the most common ones. For example, suppose you claim (x + 5)(x + 2) = x² + 10. At x = 0 both sides equal 10, so the mistake hides. At x = 3 the product is 8 × 5 = 40, but x² + 10 gives 19, so you know there is an error. (The correct answer, x² + 7x + 10, gives 40.)

A match is a good sign, not a proof. When the answer has several terms, try a second value such as 4.