Multiplying Polynomials — Definition, Formula & Examples
Multiplying polynomials is the process of finding the product of two or more polynomial expressions by distributing each term of one polynomial across every term of the other and then combining like terms.
Given two polynomials and , their product is the polynomial where each coefficient . The degree of the product equals the sum of the degrees of the factors.
Key Formula
Where:
- = Terms of the first polynomial
- = Terms of the second polynomial
How It Works
To multiply two polynomials, multiply every term in the first polynomial by every term in the second polynomial, then combine like terms. When both factors are binomials, many students use the FOIL method (First, Outer, Inner, Last) as a shortcut, but the underlying idea is always the distributive property. For larger polynomials, organize your work in rows or a grid to avoid missing any term pairs. After distributing, group terms with the same exponent and add their coefficients to simplify the result.
Worked Example
Problem: Multiply .
Step 1: Distribute across every term in the second polynomial.
Step 2: Distribute across every term in the second polynomial.
Step 3: Add the results and combine like terms.
Answer:
Another Example
Problem: Multiply .
Step 1: Apply the distributive property (FOIL): First, Outer, Inner, Last.
Step 2: Simplify each product.
Step 3: Combine like terms. The middle terms cancel.
Answer: (a difference of squares)
Visualization
Why It Matters
Multiplying polynomials is a core skill in Algebra 1 and Algebra 2 that you will use constantly when factoring expressions, solving quadratic and higher-degree equations, and simplifying rational expressions. In physics and engineering, polynomial products appear in area and volume formulas, signal processing, and modeling curved surfaces. Mastering this operation also prepares you for the Binomial Theorem, which extends polynomial multiplication to any positive integer power.
Common Mistakes
Mistake: Forgetting to multiply every term in the first polynomial by every term in the second, especially when one polynomial has three or more terms.
Correction: Use a grid or table to systematically pair each term from the first polynomial with each term from the second. The total number of partial products should equal the product of the number of terms in each factor.
Mistake: Adding exponents incorrectly or combining terms that are not like terms (e.g., adding and ).
Correction: When multiplying, add the exponents: . When combining, only add coefficients of terms with the exact same variable and exponent.