Multiplying Polynomials: FOIL Method and Special Products

Multiplying Polynomials

Introduction

Multiplying polynomials is a fundamental skill in algebra, essential for simplifying expressions and solving equations. This section covers the distributive property, the FOIL method for binomials, and special product formulas for squares and cubes.

Distributive Property

Definition and Application

• Distributive Property: For any real numbers a, b, and c, the property states that a(b + c) = ab + ac.

• This property allows you to multiply a single term by each term inside a parenthesis.

Example:

• $x^2(x - 2) = x^3 - 2x^2$

FOIL Method

Multiplying Two Binomials

The FOIL method is a shortcut for multiplying two binomials. FOIL stands for First, Outer, Inner, Last, indicating the order in which you multiply the terms.

• First: Multiply the first terms in each binomial.

• Outer: Multiply the outer terms.

• Inner: Multiply the inner terms.

• Last: Multiply the last terms in each binomial.

Example:

• $(x + 2)(x + 3)$

• First: $x \times x = x^2$

• Outer: $x \times 3 = 3x$

• Inner: $2 \times x = 2x$

• Last: $2 \times 3 = 6$

• Combine: $x^2 + 3x + 2x + 6 = x^2 + 5x + 6$

Practice Problems (FOIL)

• $(x - 5)(x - 12) = x^2 - 17x + 60$

• $(4x + 7)(-x + 6) = -4x^2 + 24x - 7x + 42 = -4x^2 + 17x + 42$

• $(x^2 - 3x)(2x + 8) = 2x^3 + 8x^2 - 6x^2 - 24x = 2x^3 + 2x^2 - 24x$

Multiplying Polynomials with More Than Two Terms

General Approach

• When multiplying polynomials with more than two terms, use the distributive property repeatedly.

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

Example:

• $(x + 4)(3x^2 - 2x + 1) = x \cdot 3x^2 + x \cdot (-2x) + x \cdot 1 + 4 \cdot 3x^2 + 4 \cdot (-2x) + 4 \cdot 1 = 3x^3 - 2x^2 + x + 12x^2 - 8x + 4 = 3x^3 + 10x^2 - 7x + 4$

Summary Table: Multiplying Polynomials

Type	Example	Result
1 Term × Many Terms	$x(x^2 - x - 2)$	$x^3 - x^2 - 2x$
2 Terms × 2 Terms (FOIL)	$(x + 3)(x - 2)$	$x^2 + x - 6$
Many Terms × Many Terms	$(x + 3)(x^2 + x - 2)$	$x^3 + 4x^2 + x - 6$

Special Products

Square Formulas

Some polynomial products follow special patterns, making them easier to compute using formulas.

• Difference of Squares: $(a + b)(a - b) = a^2 - b^2$

• Square of a Binomial: $(a + b)^2 = a^2 + 2ab + b^2$

• Square of a Binomial (Negative): $(a - b)^2 = a^2 - 2ab + b^2$

Example:

• $(x + 5)^2 = x^2 + 10x + 25$

• $(x - 6)^2 = x^2 - 12x + 36$

Practice Problems (Special Product Formulas)

• $(5x - 9)(5x + 9) = (5x)^2 - 9^2 = 25x^2 - 81$

• $(3x + 5)(3x - 5) = (3x)^2 - 5^2 = 9x^2 - 25$

Cube Formulas

Cube formulas are used for binomials raised to the third power.

• Cube of a Binomial (Positive): $(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$

• Cube of a Binomial (Negative): $(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3$

Example:

• $(x - 3)^3 = x^3 - 9x^2 + 27x - 27$

• $(2x + 4)^3 = (2x)^3 + 3(2x)^2(4) + 3(2x)(4^2) + 4^3 = 8x^3 + 48x^2 + 96x + 64$

Summary Table: Special Product Formulas

Formula	Name
$(a + b)(a - b) = a^2 - b^2$	Difference of Squares
$(a + b)^2 = a^2 + 2ab + b^2$	Square of a Binomial
$(a - b)^2 = a^2 - 2ab + b^2$	Square of a Binomial (Negative)
$(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$	Cube of a Binomial
$(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3$	Cube of a Binomial (Negative)

Key Takeaways

• Use the distributive property for multiplying any polynomials.

• Apply the FOIL method for multiplying two binomials efficiently.

• Recognize and use special product formulas to simplify calculations.

• Always combine like terms after multiplying.