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Multiplying Polynomials — Methods, Steps, and Examples

TL;DR

Multiplying polynomials means multiplying every term of one polynomial by every term of the other, then combining like terms. This article covers the distributive, FOIL, box, and vertical methods, works through six examples, and fixes the exponent and sign mistakes that lose marks.

Last updated on July 19, 2026

What Does Multiplying Polynomials Mean?

Multiplying polynomials is the operation of finding the product of two or more polynomials by multiplying each term of one by each term of the other and adding the results. A polynomial is an expression built from variables and coefficients using addition, subtraction, and whole-number powers, such as 2x² + 3x - 12.

The single governing rule is the distributive property: a(b+c) = ab + ac, applied repeatedly until every term has met every other term.

This is narrower than the general multiplication of algebraic expressions, which also covers expressions with fractional or negative powers. Polynomials keep whole-number exponents, so the term-by-term product always lands as another polynomial.

How Do You Multiply Polynomials?

Two exponent-and-coefficient rules carry every method:

• Coefficients multiply: the number parts multiply straight across, so 3×5 = 15.
• Like variables add exponents: x²×x³ = x⁵.

So 3x²×5x³ = 15x⁵: multiply 3 and 5, add the powers 2 and 3.

Is FOIL different from the distributive property? No. FOIL is just the distributive property with a memory order for the special case of two binomials; it does not extend to longer polynomials.

Methods for Multiplying Polynomials

Four named methods appear in textbooks, and all four are the distributive property wearing different layouts. Pick by the size of the polynomials, not by preference.

• Distributive method: the general rule. Multiply each term of one polynomial by every term of the other, then combine like terms. Works for any two polynomials of any length.
• FOIL (First, Outer, Inner, Last): an ordering trick for two binomials only, such as (x+4)(x+5).
• Box (grid) method: draw a grid with one polynomial's terms down the side and the other's across the top, fill each cell with a product, then add the cells.
• Vertical (column) method: stack the polynomials like a long-multiplication sum, multiply row by row, and add aligned like terms in columns.

Each method is worked in full in the examples below.

Examples of Multiplying Polynomials

Example 1

Multiply the monomials 4x³×2x². Multiply coefficients: 4×2 = 8. Add exponents of x: x³×x² = x⁵. Final answer: 8x⁵.

Example 2

Multiply (x+4)(x+5). Using the full distributive product:
(x+4)(x+5)=x⋅x+x⋅5+4⋅x+4⋅5 = x² + 9x + 20. Final answer: x² + 9x + 20.

Example 3

Multiply (2x+3)(4x+5) using FOIL. First: 2x×4x=8x²
Outer: 2x×5=10x
Inner: 3×4x=12x
Last: 3×5=15. Combine like terms to get: 8x² + 22x + 15. Final answer: 8x² + 22x + 15.

Example 4

Multiply 3y(5x+2z). Distribute 3y across both terms:
3y×5x=15xy
3y×2z=6yz. Final answer: 15xy + 6yz.

Example 5

Multiply (x+2)(x²+3x+4) using the box method. Add the cells and combine like terms: Final answer: x³ + 5x² + 10x + 8.

Example 6

Multiply (2x−3)(x²−x+5) using the vertical method. Add the two rows column by column: Final answer: 2x³ - 5x² + 13x - 15.

Properties of Multiplying Polynomials

• Closure: the product of two polynomials is always another polynomial.
• Commutative: order does not matter, P⋅Q = Q⋅P.
• Associative: grouping does not matter, (P⋅Q)⋅R = P⋅(Q⋅R).
• Distributive over addition: P⋅(Q+R) = P⋅Q + P⋅R.
• Degrees add: the degree of the product equals the sum of the degrees.

Why Multiplying Polynomials Is the Engine Behind Formulas

Polynomial multiplication exists to answer a practical question: what happens when two changing quantities combine? Area is length times width, and when both are expressions, their product is a polynomial.

Where Multiplying Polynomials Goes Sideways

Most errors are exponent slips or dropped terms. Count your terms to catch these mistakes.

Mistake 1: Multiplying exponents instead of adding them

Don't do this: x²×x³=x⁶. The correct way: Exponents of like bases add, so x²×x³=x⁵.

Mistake 2: Distributing a square across a sum

Don't do this: (x+4)²=x²+16. The correct way: Rewrite as (x+4)(x+4)=x²+8x+16.

Mistake 3: Forgetting to combine like terms

Don't do this: Stop at x²+5x+4x+20. The correct way: Collect like terms: 5x+4x=9x.

Conclusion

• Multiplying polynomials means multiplying every term of one by every term of the other, then combining like terms.
• Coefficients multiply and like-base exponents add.
• FOIL, the box method, and the vertical method are all forms of the distributive property.
• The frequent mistakes are multiplying exponents, distributing a square across a sum, and skipping the combine-like-terms step.