Polynomial Multiplication Formula

Polynomial multiplication is multiplying polynomials by distributing every term in one polynomial to every term in the other, then combining like terms.

The Formula

(x+a)(x+b)=x2+(a+b)x+ab(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)(x + 3)(x + 5), the FOIL method (First, Outer, Inner, Last) organizes the four handshakes: x⋅x+x⋅5+3⋅x+3⋅5x \cdot x + x \cdot 5 + 3 \cdot x + 3 \cdot 5.

Quick Example

(x+3)(x+5)=x2+5x+3x+15=x2+8x+15(x + 3)(x + 5) = x^2 + 5x + 3x + 15 = x^2 + 8x + 15 — distribute each term in the first factor.

Notation

FOIL: First (x⋅xx \cdot x), Outer (x⋅bx \cdot b), Inner (a⋅xa \cdot x), Last (a⋅ba \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)(x + 3)(x + 5), the FOIL method (First, Outer, Inner, Last) organizes the four handshakes: x⋅x+x⋅5+3⋅x+3⋅5x \cdot x + x \cdot 5 + 3 \cdot x + 3 \cdot 5.

Formal View

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

Worked Examples

Example 1

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

Answer

x2+7x+12x^2 + 7x + 12

First step

1
Step 1: First: x⋅x=x2x \cdot x = x^2.

Full solution

  1. 2
    Step 2: Outer: x⋅3=3xx \cdot 3 = 3x. Inner: 4⋅x=4x4 \cdot x = 4x. Last: 4⋅3=124 \cdot 3 = 12.
  2. 3
    Step 3: Combine: x2+3x+4x+12=x2+7x+12x^2 + 3x + 4x + 12 = x^2 + 7x + 12.
  3. 4
    Check: At x=1x = 1: (5)(4)=20(5)(4) = 20 and 1+7+12=201 + 7 + 12 = 20 ✓
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)(x2+3x−5)(2x - 1)(x^2 + 3x - 5).

Example 3

medium
Multiply (3x+4)(2x−5)(3x + 4)(2x - 5) using FOIL.

Common Mistakes

  • Squaring a binomial as a2+b2a^2+b^2 - (a+b)2=a2+2ab+b2(a+b)^2=a^2+2ab+b^2 includes the cross term.
  • Missing a cross-product term (forgetting Outer or Inner in FOIL) - every term must meet every term.
  • Adding exponents wrong when multiplying terms - x2⋅x3=x5x^2\cdot x^3=x^5 (add exponents only when MULTIPLYING).

Why This Formula Matters

It builds the higher-degree polynomials algebra runs on and is the exact reverse of factoring, so mastering it makes factoring legible. Forgetting a cross term (a missed handshake) is the most common multiplication error. Recognizing it by "Am I multiplying expressions so that every term in one meets every term in the other?" — rather than by familiar numbers — is what lets a student tell it apart from polynomial addition and factoring and distributive property (single term) in a mixed problem set.

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)(x + 3)(x + 5), the FOIL method (First, Outer, Inner, Last) organizes the four handshakes: x⋅x+x⋅5+3⋅x+3⋅5x \cdot x + x \cdot 5 + 3 \cdot x + 3 \cdot 5.

What do the symbols mean in the Polynomial Multiplication formula?

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

Why is the Polynomial Multiplication formula important in Math?

It builds the higher-degree polynomials algebra runs on and is the exact reverse of factoring, so mastering it makes factoring legible. Forgetting a cross term (a missed handshake) is the most common multiplication error. Recognizing it by "Am I multiplying expressions so that every term in one meets every term in the other?" — rather than by familiar numbers — is what lets a student tell it apart from polynomial addition and factoring and distributive property (single term) in a mixed problem set.

What do students get wrong about Polynomial Multiplication?

The procedure for polynomial multiplication is the easy part; the trap is squaring a binomial as a2+b2a^2+b^2. Asking "Am I multiplying expressions so that every term in one meets every term in the other?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

What should I learn before the Polynomial Multiplication formula?

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

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