Polynomial Multiplication Formula

The Formula

(x + a)(x + b) = x^2 + (a + b)x + ab

When to use: 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.

Quick Example

(x + 3)(x + 5) = x^2 + 5x + 3x + 15 = x^2 + 8x + 15 — distribute each term in the first factor.

Notation

FOIL: First (x \cdot x), Outer (x \cdot b), Inner (a \cdot x), Last (a \cdot b). For larger polynomials, multiply each term by each term.

What This Formula Means

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.

Formal View

For P(x) = \sum_{i=0}^{m} a_i x^i and Q(x) = \sum_{j=0}^{n} b_j x^j: (PQ)(x) = \sum_{k=0}^{m+n} c_k x^k where c_k = \sum_{i+j=k} a_i b_j. Note \deg(PQ) = \deg(P) + \deg(Q).

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).

Common Mistakes

  • Only multiplying the first terms and last terms, missing the middle (cross) terms
  • Forgetting to combine like terms after distributing
  • Errors with signs when multiplying negative terms: (-3)(+2x) = -6x, not +6x

Why This Formula Matters

Required for expanding expressions, deriving identities like difference of squares, and understanding factoring as the reverse process.

Frequently Asked Questions

What is the Polynomial Multiplication formula?

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

How do you use the Polynomial Multiplication formula?

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.

What do the symbols mean in the Polynomial Multiplication formula?

FOIL: First (x \cdot x), Outer (x \cdot b), Inner (a \cdot x), Last (a \cdot b). For larger polynomials, multiply each term by each term.

Why is the Polynomial Multiplication formula important in Math?

Required for expanding expressions, deriving identities like difference of squares, and understanding factoring as the reverse process.

What do students get wrong about Polynomial Multiplication?

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

What should I learn before the Polynomial Multiplication formula?

Before studying the Polynomial Multiplication formula, you should understand: polynomials, distributive property.

Want the Full Guide?

This formula is covered in depth in our complete guide:

Polynomial Long Division: Step-by-Step Method with Examples →
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