Multiplying polynomials

Multiplying monomials

Multiply coefficients, then add exponents of the same base (using the exponent laws).

• 3x² × 4x⁵ = 12x⁷
• (−2ab)(5a²b³) = −10a³b⁴

Monomial × polynomial

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

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

Binomial × binomial

Each term in the first binomial multiplies each term in the second. FOIL helps you remember the four products:

Add and combine: x² + 8x + 15

The area model (box method)

Add all the boxes: x² + 8x + 15. This works for any size of polynomial and keeps track of every product.

With negatives

Example: (2x − 3)(x + 4)

• F: 2x²
• O: 8x
• I: −3x
• L: −12

Result: 2x² + 5x − 12

Special products

• (x + 6)² = x² + 12x + 36
• (3y − 1)(3y + 1) = 9y² − 1

Binomial × trinomial

Multiply each term of the binomial by every term of the trinomial: 2 × 3 = 6 products.

Example: (x + 2)(x² − 3x + 1)

• x(x² − 3x + 1) = x³ − 3x² + x
• 2(x² − 3x + 1) = 2x² − 6x + 2
• Sum: x³ − x² − 5x + 2

Checking

Substitute a value. For (x + 3)(x + 5) with x = 1: 4 × 6 = 24; x² + 8x + 15 = 1 + 8 + 15 = 24 ✓.

Worked examples

Example 1: −4y(y² − 2y + 3) → −4y³ + 8y² − 12y.

Example 2: (x − 7)(x + 2) → x² − 5x − 14.

Example 3: (2a + 5)² → 4a² + 20a + 25.

Common mistakes

• (x + 3)² = x² + 9 (missing the middle term 6x).
• Forgetting to distribute to every term.
• Multiplying exponents instead of adding: x² × x³ = x⁵, not x⁶.

Try it yourself

1. 5x²(3x − 2)
2. (x + 4)(x + 9)
3. (3x − 2)(2x + 5)
4. (x − 5)²
5. (y + 1)(y² + 2y − 3)

Answers: 1) 15x³ − 10x² 2) x² + 13x + 36 3) 6x² + 11x − 10 4) x² − 10x + 25 5) y³ + 3y² − y − 3

Quick recap

• Multiply every term by every term (distributive property).
• Use FOIL or the box method for binomials.
• Add exponents, combine like terms, and check by substituting.