Multiplying Polynomials (FOIL & Distributive)

TipLearning Objectives

By the end of this lesson, you’ll be able to:

  • Multiply binomials and larger polynomials confidently.
  • Apply the FOIL pattern when multiplying binomials.
  • Use the distributive property to expand any polynomial expression.
  • Combine like terms to write expressions in simplified form.

Key Ideas

  • Distributive property \[ a(b + c) = ab + ac \]

  • FOIL for binomials
    Multiply:

    • First terms
    • Outer terms
    • Inner terms
    • Last terms

  • General polynomial multiplication
    Multiply each term in the first polynomial by each term in the second, then combine like terms.

Common Problem Types

1. Binomial × Binomial

Use FOIL as a structured way to distribute.

2. Binomial × Polynomial

Distribute each binomial term across the entire second polynomial.

3. Polynomial × Polynomial

Multiply term-by-term; expect multiple like terms to combine.

4. Special Cases

Perfect square patterns such as \((a - b)^2\) or \((a + b)^2\).

Strategies

  • Start by distributing systematically—FOIL is just a special case of distribution.
  • Keep your work organized by lining up like terms.
  • Multiply coefficients and variables separately:
    \[ (ax^m)(bx^n) = abx^{m+n} \]
  • After expanding, always combine like terms.
  • Rewrite your final answer in descending power order.

Worked Examples

Example 1 — FOIL

Multiply:
\[ (x + 3)(x + 5) \]

Solution:
Apply FOIL:

\[ \begin{split} \text{First: } & x \cdot x = x^2 \\ \text{Outer: } & x \cdot 5 = 5x \\ \text{Inner: } & 3 \cdot x = 3x \\ \text{Last: } & 3 \cdot 5 = 15 \end{split} \]

Combine like terms: \[ x^2 + 8x + 15 \]

Example 2 — Full Distribution

Multiply:
\[ (2x - 1)(3x^2 + x + 4) \]

Solution:
Distribute each term in \((2x - 1)\) across \((3x^2 + x + 4)\).

\[ \begin{split} 2x(3x^2 + x + 4) &= 6x^3 + 2x^2 + 8x \\ -1(3x^2 + x + 4) &= -3x^2 - x - 4 \end{split} \]

Combine: \[ 6x^3 + x^2 + 6x - 4 \]

WarningCommon Mistakes
  • Forgetting one of the FOIL products when multiplying binomials.
  • Not distributing to every term in the second polynomial.
  • Leaving the result uncombined—always merge like terms.

Practice Problems

  1. \((x + 2)(x + 7)\)
  2. \((2x - 3)(x - 4)\)
  3. \((x + 5)(x^2 - x + 3)\)
  4. \((3y - 1)(2y + 4)\)
  5. \((a - 3)(a - 3)\)

1.
\[ x^2 + 9x + 14 \]

2.
\[ 2x^2 - 11x + 12 \]

3.
\[ x^3 + 4x^2 + 2x + 15 \]

4.
\[ 6y^2 + 10y - 4 \]

5.
\[ a^2 - 6a + 9 \]

Summary

  • Multiply polynomials using distribution (FOIL is just one case).
  • Multiply each term in one polynomial by each term in the other.
  • Combine like terms after expanding.
  • Keep expressions in descending powers for clarity.
  • Stay organized to avoid missing any products.
  • FOIL is simply the distributive property applied to binomials.
  • Write out all products before combining—organization prevents mistakes.
  • Combine like terms carefully after multiplying.
  • Use descending order to present your final answer cleanly.