Multiplying Polynomials

Understanding how to multiply binomials and trinomials

Multiplying Two Binomials

A binomial is a polynomial with two terms, like (a + b).

FOIL Method (First, Outer, Inner, Last)

FOIL is a mnemonic for the standard distribution method.

Example: Multiply (x + 2)(x + 5)

1. First: x * x = x²

2. Outer: x * 5 = 5x

3. Inner: 2 * x = 2x

4. Last: 2 * 5 = 10

(x + 2)(x + 5) = x² + 5x + 2x + 10 = x² + 7x + 10

Box Method

This visual method ensures you don't miss any terms.

Example: Multiply (x + 2)(x + 5)

1. Create a 2×2 box with terms on each axis

2. Multiply to fill each cell:

x * x = x² | x * 5 = 5x
2 * x = 2x | 2 * 5 = 10

3. Combine all terms: x² + 5x + 2x + 10

4. Simplify: x² + 7x + 10

Multiplying Binomials with Trinomials

A trinomial has three terms, like (a + b + c).

Distributive Property

Multiply each term in the first polynomial by each term in the second.

Example: Multiply (x - 3)(x² + 2x - 4)

1. Distribute x across the trinomial:

x(x² + 2x - 4) = x³ + 2x² - 4x

2. Distribute -3 across the trinomial:

-3(x² + 2x - 4) = -3x² - 6x + 12

3. Combine all terms:

x³ + 2x² - 4x - 3x² - 6x + 12

4. Simplify by combining like terms:

x³ - x² - 10x + 12

Box Method for Larger Polynomials

The box method scales well for more complex multiplications.

Example: Multiply (x - 3)(x² + 2x - 4)

1. Create a 2×3 box with terms on each axis

2. Multiply to fill each cell:

x * x² = x³ | x * 2x = 2x² | x * (-4) = -4x
-3 * x² = -3x² | -3 * 2x = -6x | -3 * (-4) = 12

3. Combine all terms: x³ + 2x² - 4x - 3x² - 6x + 12

4. Simplify: x³ - x² - 10x + 12

Key Points to Remember

• Always multiply every term in the first polynomial by every term in the second polynomial
• Pay close attention to signs - this is where most mistakes happen
• Combine like terms after distribution
• The Box Method helps visualize the process and prevents missing terms
• For (binomial)×(trinomial), you should have 2×3=6 terms before simplification
• Practice with different coefficients and signs to build confidence

Math Learning Resources | Polynomial Multiplication