Multiplying Polynomials

Introduction to Polynomial Multiplication

Unlike addition and subtraction where we combine like terms, multiplication creates new terms by multiplying each term in one polynomial by each term in the other.

Key Principle: The distributive property a(b + c) = ab + ac

This extends to polynomials of any size!

Multiplying Monomials

A monomial is a single term.

Rules:

1. Multiply coefficients
2. Add exponents of like bases (product rule)

Example 1: 3x · 5x = (3 · 5)(x · x) = 15x²

Example 2: (2x²)(4x³) = (2 · 4)(x² · x³) = 8x⁵

Example 3: (3x²y)(5xy³) = (3 · 5)(x² · x)(y · y³) = 15x³y⁴

Example 4: (-4a²b)(3ab³) = (-4 · 3)(a² · a)(b · b³) = -12a³b⁴

Multiplying Monomial by Polynomial

Use the distributive property: multiply the monomial by each term.

Example 1: 3(x + 5) = 3x + 15

Example 2: 2x(3x + 4) = 2x · 3x + 2x · 4 = 6x² + 8x

Example 3: 5x²(2x² - 3x + 4) = 5x² · 2x² + 5x² · (-3x) + 5x² · 4 = 10x⁴ - 15x³ + 20x²

Example 4: -3x(4x² - 2x + 5) = -3x · 4x² + (-3x) · (-2x) + (-3x) · 5 = -12x³ + 6x² - 15x

Example 5: 2xy(3x²y - 4xy + y²) = 6x³y² - 8x²y² + 2xy³

Multiplying Binomials (FOIL Method)

FOIL stands for: First, Outer, Inner, Last

For (a + b)(c + d):

• First: a · c
• Outer: a · d
• Inner: b · c
• Last: b · d

Then combine like terms.

Example 1: (x + 3)(x + 5)

F: x · x = x² O: x · 5 = 5x I: 3 · x = 3x L: 3 · 5 = 15

Combine: x² + 5x + 3x + 15 = x² + 8x + 15

Example 2: (x + 4)(x - 2)

F: x · x = x² O: x · (-2) = -2x I: 4 · x = 4x L: 4 · (-2) = -8

Combine: x² - 2x + 4x - 8 = x² + 2x - 8

Example 3: (2x + 3)(x + 5)

F: 2x · x = 2x² O: 2x · 5 = 10x I: 3 · x = 3x L: 3 · 5 = 15

Combine: 2x² + 10x + 3x + 15 = 2x² + 13x + 15

Example 4: (3x - 2)(2x - 5)

F: 3x · 2x = 6x² O: 3x · (-5) = -15x I: (-2) · 2x = -4x L: (-2) · (-5) = 10

Combine: 6x² - 15x - 4x + 10 = 6x² - 19x + 10

Alternative: Box Method for Binomials

Draw a 2×2 grid and multiply.

Example: (x + 3)(x + 5)

Create a box with:

• Top row: x and 5
• Left column: x and 3
• Fill boxes: x·x=x², x·5=5x, 3·x=3x, 3·5=15

Sum all boxes: x² + 5x + 3x + 15 = x² + 8x + 15

Special Products: Difference of Squares

(a + b)(a - b) = a² - b²

The outer and inner terms cancel!

Example 1: (x + 5)(x - 5) = x² - 25

Example 2: (3x + 4)(3x - 4) = (3x)² - 4² = 9x² - 16

Example 3: (2x + 7)(2x - 7) = 4x² - 49

Why it works: (x + 5)(x - 5) = x² - 5x + 5x - 25 = x² - 25 (middle terms cancel)

Special Products: Perfect Square Trinomials

(a + b)² = a² + 2ab + b² (a - b)² = a² - 2ab + b²

Example 1: (x + 3)² = x² + 2(x)(3) + 3² = x² + 6x + 9

Example 2: (x - 5)² = x² - 2(x)(5) + 5² = x² - 10x + 25

Example 3: (2x + 4)² = (2x)² + 2(2x)(4) + 4² = 4x² + 16x + 16

Example 4: (3x - 2)² = (3x)² - 2(3x)(2) + 2² = 9x² - 12x + 4

Common Mistake: (x + 3)² ≠ x² + 9 You MUST include the middle term: x² + 6x + 9

Multiplying Binomial by Trinomial

Distribute each term of the binomial to all terms of the trinomial.

Example 1: (x + 2)(x² + 3x + 4)

x(x² + 3x + 4) + 2(x² + 3x + 4) = x³ + 3x² + 4x + 2x² + 6x + 8 = x³ + 5x² + 10x + 8

Example 2: (2x - 1)(x² - x + 3)

2x(x² - x + 3) - 1(x² - x + 3) = 2x³ - 2x² + 6x - x² + x - 3 = 2x³ - 3x² + 7x - 3

Example 3: (x + 3)(2x² - 5x + 1)

= x(2x² - 5x + 1) + 3(2x² - 5x + 1) = 2x³ - 5x² + x + 6x² - 15x + 3 = 2x³ + x² - 14x + 3

Box Method for Larger Polynomials

Useful for trinomial × trinomial or larger.

Example: (x + 2)(x² + 3x + 4)

Create a box with x² 3x and 4 across the top, x and 2 down the side:

• x times x² = x³
• x times 3x = 3x²
• x times 4 = 4x
• 2 times x² = 2x²
• 2 times 3x = 6x
• 2 times 4 = 8

Sum: x³ + 3x² + 4x + 2x² + 6x + 8 = x³ + 5x² + 10x + 8

Multiplying Trinomials

Distribute systematically - each term times each term.

Example: (x + 1)(x + 2)(x + 3)

First multiply (x + 1)(x + 2): = x² + 2x + x + 2 = x² + 3x + 2

Then multiply result by (x + 3): (x² + 3x + 2)(x + 3) = x(x² + 3x + 2) + 3(x² + 3x + 2) = x³ + 3x² + 2x + 3x² + 9x + 6 = x³ + 6x² + 11x + 6

Polynomials with Multiple Variables

Same rules apply!

Example 1: (x + y)(x + 2y)

F: x · x = x² O: x · 2y = 2xy I: y · x = xy L: y · 2y = 2y²

Result: x² + 2xy + xy + 2y² = x² + 3xy + 2y²

Example 2: (2a + b)(3a - 2b)

= 6a² - 4ab + 3ab - 2b² = 6a² - ab - 2b²

Example 3: (x + y)² = x² + 2xy + y²

Common Mistakes to Avoid

1. Forgetting the middle term in perfect squares Wrong: (x + 3)² = x² + 9 Right: (x + 3)² = x² + 6x + 9

2. Sign errors with negatives Careful: (-3)(-2) = +6, not -6

3. Not combining like terms After FOIL, always combine!

4. Forgetting to distribute to ALL terms x(x² + 2x + 1) has THREE terms to multiply

5. Exponent errors x · x = x², not 2x or x

Applications: Area Problems

Example 1: Rectangle with length (x + 5) and width (x + 3). Find area.

Area = length × width = (x + 5)(x + 3) = x² + 3x + 5x + 15 = x² + 8x + 15

Example 2: Square with side length (2x + 1). Find area.

Area = (side)² = (2x + 1)² = 4x² + 4x + 1

Volume Applications

Example: Box with dimensions (x + 2), (x + 1), and x. Find volume.

V = length × width × height = (x + 2)(x + 1)(x)

First: (x + 2)(x + 1) = x² + 3x + 2 Then: (x² + 3x + 2)(x) = x³ + 3x² + 2x

Patterns and Shortcuts

Binomial Squared: (a ± b)² = a² ± 2ab + b²

Difference of Squares: (a + b)(a - b) = a² - b²

Sum and Difference: If you see (x + k)(x - k), answer is x² - k²

Trinomial × Monomial: Distribute the monomial to each term

Simplifying Complex Products

Example: 2x(x + 3)(x - 2)

Method 1 - Left to right: 2x(x + 3) = 2x² + 6x (2x² + 6x)(x - 2) = 2x³ - 4x² + 6x² - 12x = 2x³ + 2x² - 12x

Method 2 - Binomials first: (x + 3)(x - 2) = x² + x - 6 2x(x² + x - 6) = 2x³ + 2x² - 12x

Same answer either way!

Word Problem

Example: The profit from selling x items is (20x - 100) dollars. If you sell (x + 5) batches, what is the total profit?

Total profit = (20x - 100)(x + 5) = 20x² + 100x - 100x - 500 = 20x² - 500

Checking Your Work

Method 1: Substitute x = 1 Evaluate both expressions with x = 1.

Example: Does (x + 2)(x + 3) = x² + 5x + 6? Left: (1 + 2)(1 + 3) = 3 · 4 = 12 Right: 1² + 5(1) + 6 = 1 + 5 + 6 = 12 ✓

Method 2: Factor your answer If you can factor back to the original, it's correct!

Quick Reference

| Product Type | Pattern | Example | |--------------|---------|---------| | Monomial × Monomial | Multiply coefficients, add exponents | 3x · 5x = 15x² | | Monomial × Polynomial | Distribute | 2x(x + 3) = 2x² + 6x | | Binomial × Binomial | FOIL | (x+2)(x+3) = x²+5x+6 | | (a+b)(a-b) | a² - b² | (x+5)(x-5) = x²-25 | | (a±b)² | a² ± 2ab + b² | (x+3)² = x²+6x+9 |

Practice Strategy

Level 1: Monomials

• 2x · 3x
• 5x² · 4x³

Level 2: Distribute

• 3(x + 4)
• 2x(3x - 5)

Level 3: FOIL

• (x + 3)(x + 4)
• (x - 2)(x + 5)

Level 4: Special products

• (x + 5)(x - 5)
• (x + 3)²

Level 5: Complex

• (2x + 3)(x² - x + 1)
• (x + 1)(x + 2)(x + 3)

Tips for Success

• Write neatly and organize work
• Show all steps - don't skip FOIL terms
• Combine like terms at the end
• Check signs carefully
• Practice special products until automatic
• Use box method for complex problems
• Always verify your answer