10.3 Multiply Polynomials

Multiplying Polynomials

Multiplying polynomials lets you combine expressions with variables and constants into a single, simplified expression. This skill builds directly on the distributive property you already know from arithmetic, and it's essential for working with areas, volumes, and more complex algebra problems later on.

You'll use three main approaches here, depending on what you're multiplying: distributing a monomial, the FOIL method for two binomials, and the general method for larger polynomials.

Algebraic Expressions and Components

A polynomial is an algebraic expression made up of terms, where each term has a variable raised to a whole-number exponent, multiplied by a constant (called a coefficient). For example, $3x^2 + 4x - 5$ has three terms.

A few quick definitions to keep straight:

• Variables are letters (like $x$) representing unknown quantities
• Constants are plain numbers with no variable attached (like $-5$)
• Like terms share the same variable raised to the same power ($3x^2$ and $7x^2$ are like terms; $3x^2$ and $3x$ are not)

When you multiply polynomials, you always get a new polynomial. The final step is always to combine like terms to simplify your answer.

Distribution for Polynomial Multiplication

The distributive property says that $a(b + c) = ab + ac$. When you multiply a monomial (a single term) by a polynomial, you distribute that monomial to every term inside the polynomial.

Here's the process:

1. Multiply the monomial by the first term of the polynomial
2. Multiply the monomial by the second term
3. Continue for every term in the polynomial
4. Write all the results as one expression (combine like terms if needed)

Example: $2x(3x^2 + 4x - 5)$

• $2x \cdot 3x^2 = 6x^3$
• $2x \cdot 4x = 8x^2$
• $2x \cdot (-5) = -10x$
• Final answer: $6x^3 + 8x^2 - 10x$

Notice how the exponents work: when you multiply $2x \cdot 3x^2$, you multiply the coefficients ($2 \cdot 3 = 6$) and add the exponents on $x$ ($1 + 2 = 3$).

FOIL Method for Binomials

FOIL stands for First, Outer, Inner, Last. It's a shortcut specifically for multiplying two binomials (expressions with exactly two terms each).

Given $(a + b)(c + d)$, follow these four steps:

1. First: Multiply the first terms of each binomial → $a \cdot c$
2. Outer: Multiply the outermost terms → $a \cdot d$
3. Inner: Multiply the innermost terms → $b \cdot c$
4. Last: Multiply the last terms of each binomial → $b \cdot d$

Then add all four results and combine like terms: $ac + ad + bc + bd$

Example: $(2x + 3)(x - 4)$

1. First: $2x \cdot x = 2x^2$
2. Outer: $2x \cdot (-4) = -8x$
3. Inner: $3 \cdot x = 3x$
4. Last: $3 \cdot (-4) = -12$

Now combine like terms: the middle terms $-8x$ and $3x$ are like terms, so $-8x + 3x = -5x$.

Final answer: $2x^2 - 5x - 12$

FOIL is really just the distributive property applied twice. It's a handy pattern to memorize, but it only works for two binomials. For anything bigger, use the general method below.

Expansion of Trinomial-Binomial Products

When one (or both) of your polynomials has more than two terms, FOIL won't cover it. Instead, distribute each term of one polynomial across every term of the other. Think of it as doing the distributive property multiple times in a row.

Steps:

1. Pick one polynomial (usually the longer one) and go term by term
2. Multiply each of its terms by every term in the other polynomial
3. Write out all the products
4. Combine like terms

Example: $(2x^2 + 3x - 1)(x + 2)$

Distribute each term of the trinomial across $(x + 2)$:

• $2x^2 \cdot x = 2x^3$ and $2x^2 \cdot 2 = 4x^2$
• $3x \cdot x = 3x^2$ and $3x \cdot 2 = 6x$
• $-1 \cdot x = -x$ and $-1 \cdot 2 = -2$

Now collect everything: $2x^3 + 4x^2 + 3x^2 + 6x - x - 2$

Combine like terms: $4x^2 + 3x^2 = 7x^2$ and $6x + (-x) = 5x$

Final answer: $2x^3 + 7x^2 + 5x - 2$

A good habit: after combining like terms, check that your answer is written in descending order of exponents (highest power first). This makes it easier to read and is the standard way to present a polynomial.