5.3 Multiply Polynomials

Multiplying Polynomials

Multiplying polynomials is about combining simpler expressions into more complex ones. Every technique here builds on one core idea: the distributive property. Whether you're multiplying monomials, using FOIL on binomials, or expanding larger products, you're always distributing and then combining like terms.

Special product forms give you shortcuts for common patterns. Recognizing these will save time on exams and set you up for factoring later.

Multiplying Polynomials

Multiplication of Monomials

A monomial is a single term (like $3x^2$ or $-5y$). To multiply two monomials:

1. Multiply the coefficients (the number parts)
2. Add the exponents of matching variables, using the product rule: $x^a \cdot x^b = x^{a+b}$

For example: $3x^2 \cdot 4x^3 = (3 \cdot 4)(x^{2+3}) = 12x^5$

If you have multiple variables, handle each one separately. With $2x^3y \cdot 5x^2y^4$, multiply the coefficients ($2 \cdot 5 = 10$), add the $x$ exponents ($3 + 2 = 5$), and add the $y$ exponents ($1 + 4 = 5$). The result is $10x^5y^5$.

Polynomial by Monomial Multiplication

This is just the distributive property in action. You multiply the monomial by each term in the polynomial separately.

Example: $3x^2(2x^3 + 4x - 1)$

1. $3x^2 \cdot 2x^3 = 6x^5$
2. $3x^2 \cdot 4x = 12x^3$
3. $3x^2 \cdot (-1) = -3x^2$

Result: $6x^5 + 12x^3 - 3x^2$

A common mistake is forgetting to distribute to every term, especially the constant at the end. Don't skip that last term.

Binomial Multiplication (FOIL)

When you multiply two binomials, the FOIL method keeps you organized. FOIL stands for First, Outer, Inner, Last.

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

1. First: $x \cdot x = x^2$
2. Outer: $x \cdot (-2) = -2x$
3. Inner: $3 \cdot x = 3x$
4. Last: $3 \cdot (-2) = -6$

Combine like terms: $x^2 - 2x + 3x - 6 = x^2 + x - 6$

FOIL is really just a specific case of distribution. It only works for two binomials. For anything larger, you'll need the general method below.

Multiplication of Larger Polynomials

For polynomials with more than two terms, distribute every term in the first polynomial to every term in the second. Then combine like terms.

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

1. Distribute $2x^2$: $2x^2 \cdot x + 2x^2 \cdot 2 = 2x^3 + 4x^2$
2. Distribute $3x$: $3x \cdot x + 3x \cdot 2 = 3x^2 + 6x$
3. Distribute $-1$: $-1 \cdot x + (-1) \cdot 2 = -x - 2$

Now combine like terms:

$2x^3 + 4x^2 + 3x^2 + 6x - x - 2 = 2x^3 + 7x^2 + 5x - 2$

A helpful check: if the first polynomial has 3 terms and the second has 2 terms, you should get $3 \times 2 = 6$ individual products before combining. Count them to make sure you haven't missed any.

Special Product Forms

These patterns show up constantly. Memorizing them will speed up both multiplication and factoring.

• Perfect square binomials: $(a + b)^2 = a^2 + 2ab + b^2$ $(a - b)^2 = a^2 - 2ab + b^2$

  Example: $(x + 3)^2 = x^2 + 2(x)(3) + 9 = x^2 + 6x + 9$

  The most common error here is writing $(x + 3)^2 = x^2 + 9$. Don't forget the middle term $2ab$.

• Difference of squares: $(a + b)(a - b) = a^2 - b^2$

  Example: $(x + 3)(x - 3) = x^2 - 9$

  The middle terms cancel out, which is why there's no middle term in the result.

• Sum of cubes: $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$

• Difference of cubes: $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$ A memory trick for cubes: the binomial factor matches the sign of the original expression, the $ab$ term in the trinomial takes the opposite sign, and the last term is always positive.

Products of Polynomial Functions

When you're given two polynomial functions and asked to evaluate their product at a specific value, you have two approaches:

Approach 1: Evaluate, then multiply. Plug the value into each function separately, then multiply the results.

If $f(x) = 2x + 1$ and $g(x) = x - 3$, find $f(2) \cdot g(2)$:

1. $f(2) = 2(2) + 1 = 5$

2. $g(2) = 2 - 3 = -1$

3. $f(2) \cdot g(2) = 5 \cdot (-1) = -5$

Approach 2: Multiply first, then evaluate. Find the product $(f \cdot g)(x)$ by multiplying the polynomials, then substitute.

1. $(f \cdot g)(x) = (2x + 1)(x - 3) = 2x^2 - 6x + x - 3 = 2x^2 - 5x - 3$

2. $(f \cdot g)(2) = 2(4) - 5(2) - 3 = 8 - 10 - 3 = -5$

Both approaches give the same answer. Approach 1 is usually faster for a single value. Approach 2 is better when you need the product function itself or need to evaluate at multiple values.

Understanding Polynomial Structure

A few terms worth knowing as you work with polynomial products:

• Degree: The highest power of the variable. In $2x^3 + 7x^2 + 5x - 2$, the degree is 3. When you multiply two polynomials, the degree of the product equals the sum of their degrees.
• Constant term: The term with no variable. In the example above, it's $-2$.
• Factoring is the reverse of multiplication. You're taking a polynomial and breaking it back into the factors that were multiplied together. Everything you practice here with multiplication directly prepares you for factoring in upcoming sections.