Polynomial Multiplication Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Polynomial Multiplication.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Multiplying polynomials by distributing every term in one polynomial to every term in the other, then combining like terms.

Each term in the first polynomial must 'shake hands' with every term in the second. For two binomials like (x + 3)(x + 5), the FOIL method (First, Outer, Inner, Last) organizes the four handshakes: x \cdot x + x \cdot 5 + 3 \cdot x + 3 \cdot 5.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Polynomial multiplication is repeated distribution—multiply each term by each term, then combine like terms.

Common stuck point: FOIL only works for two binomials. For larger polynomials, use systematic distribution (every term times every term).

Sense of Study hint: Set up a multiplication grid with one polynomial across the top and the other down the side, then fill in every cell.

Worked Examples

Example 1

easy
Multiply (x + 4)(x + 3) using FOIL.

Solution

  1. 1
    Step 1: First: x \cdot x = x^2.
  2. 2
    Step 2: Outer: x \cdot 3 = 3x. Inner: 4 \cdot x = 4x. Last: 4 \cdot 3 = 12.
  3. 3
    Step 3: Combine: x^2 + 3x + 4x + 12 = x^2 + 7x + 12.
  4. 4
    Check: At x = 1: (5)(4) = 20 and 1 + 7 + 12 = 20 ✓

Answer

x^2 + 7x + 12
FOIL (First, Outer, Inner, Last) is a mnemonic for multiplying two binomials. Each term in the first binomial multiplies each term in the second.

Example 2

medium
Expand (2x - 1)(x^2 + 3x - 5).

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Multiply (x - 2)(x + 6).

Example 2

hard
Expand (x + 1)^2(x - 1).
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Background Knowledge

These ideas may be useful before you work through the harder examples.

polynomialsdistributive property