Multiplying Polynomials

Multiplying Monomials

Monomial: A single term (like 3x or 5x²)

To multiply: Multiply coefficients, add exponents

Rule: x^a × x^b = x^(a+b)

Example 1: Basic Multiplication

(3x)(4x)

Multiply coefficients: 3 × 4 = 12 Add exponents: x¹ × x¹ = x²

Answer: 12x²

Example 2: With Exponents

(2x³)(5x²)

Coefficients: 2 × 5 = 10 Exponents: x³ × x² = x⁵

Answer: 10x⁵

Example 3: Negative Coefficients

(−3x²)(4x⁴)

Coefficients: −3 × 4 = −12 Exponents: x² × x⁴ = x⁶

Answer: −12x⁶

Multiplying Monomial by Polynomial

Use distributive property: Multiply monomial by each term

a(b + c) = ab + ac

Example 1: Monomial × Binomial

3x(2x + 5)

Distribute:

• 3x × 2x = 6x²
• 3x × 5 = 15x

Answer: 6x² + 15x

Example 2: Negative Monomial

−2x(3x² − 4x + 1)

Distribute:

• −2x × 3x² = −6x³
• −2x × (−4x) = 8x²
• −2x × 1 = −2x

Answer: −6x³ + 8x² − 2x

Multiplying Binomials

Binomial: Two terms (like x + 3)

Method 1: Distributive Property (twice) Method 2: FOIL

FOIL Method

FOIL stands for:

• First terms
• Outer terms
• Inner terms
• Last terms

Example 1: Using FOIL

(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

Answer: x² + 8x + 15

Example 2: With Subtraction

(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

Answer: x² − 2x − 8

Example 3: Both Negative

(x − 5)(x − 3)

F: x × x = x² O: x × (−3) = −3x I: −5 × x = −5x L: −5 × (−3) = 15

Combine: x² − 3x − 5x + 15

Answer: x² − 8x + 15

Example 4: With Coefficients

(2x + 1)(3x − 4)

F: 2x × 3x = 6x² O: 2x × (−4) = −8x I: 1 × 3x = 3x L: 1 × (−4) = −4

Combine: 6x² − 8x + 3x − 4

Answer: 6x² − 5x − 4

Using Distributive Property

Alternative to FOIL: Distribute each term of first to second

Example: (x + 2)(x + 7)

Distribute x:

• x(x + 7) = x² + 7x

Distribute 2:

• 2(x + 7) = 2x + 14

Add results:

• x² + 7x + 2x + 14
• x² + 9x + 14

Answer: x² + 9x + 14

Special Products

Memorize these patterns for speed!

Pattern 1: Perfect Square (a + b)²

(a + b)² = a² + 2ab + b²

Example: (x + 5)²

• = x² + 2(x)(5) + 5²
• = x² + 10x + 25

Pattern 2: Perfect Square (a − b)²

(a − b)² = a² − 2ab + b²

Example: (x − 3)²

• = x² − 2(x)(3) + 3²
• = x² − 6x + 9

Pattern 3: Difference of Squares

(a + b)(a − b) = a² − b²

Example: (x + 4)(x − 4)

• = x² − 4²
• = x² − 16

No middle term!

Multiplying Trinomial by Binomial

Use distributive property for each term

Example: (x + 2)(x² + 3x − 1)

Distribute x:

• x(x² + 3x − 1) = x³ + 3x² − x

Distribute 2:

• 2(x² + 3x − 1) = 2x² + 6x − 2

Add:

• x³ + 3x² − x + 2x² + 6x − 2
• x³ + 5x² + 5x − 2

Answer: x³ + 5x² + 5x − 2

Common Mistakes

❌ (x + 3)² = x² + 9

• Wrong! Must use (a + b)² = a² + 2ab + b²
• ✓ (x + 3)² = x² + 6x + 9

❌ Forgetting negative signs

• (x − 2)(x + 5): Remember −2 × 5 = −10

❌ Not combining like terms

• Must simplify final answer!

Vertical Multiplication

Alternative method: Stack and multiply like numbers

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

      x + 3
  ×  2x + 5
  _________
     5x + 15   (multiply by 5)
  2x² + 6x     (multiply by 2x)
  ___________
  2x² + 11x + 15

Answer: 2x² + 11x + 15

Real-World Applications

Area: Rectangle with sides (x + 3) and (x + 5)

• Area = (x + 3)(x + 5) = x² + 8x + 15

Revenue: Price (10 − x) × Quantity (50 + 2x)

• R = (10 − x)(50 + 2x) = 500 + 20x − 50x − 2x²

Volume: Box dimensions (x)(x + 2)(x − 1)

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

Practice