Multiplying Polynomials

Overview

Multiplying polynomials involves using the distributive property to multiply each term in one polynomial by each term in the other. When multiplying polynomials, be careful to combine like terms after distributing.

Steps to Multiply Polynomials

  1. Multiply each term in the first polynomial by each term in the second polynomial.
  2. Combine like terms, if any.
  3. Simplify the expression.

Example 1: Multiplying a Monomial and a Polynomial

Consider the expressions:

\( 3x \cdot (2x^2 + 4x - 5) \)

Distribute \( 3x \) to each term:

\( (3x \cdot 2x^2) + (3x \cdot 4x) + (3x \cdot -5) \) \( = 6x^3 + 12x^2 - 15x \)

The result is:

\( 6x^3 + 12x^2 - 15x \)

Example 2: Multiplying Two Binomials

Consider the binomials:

\( (x + 3)(x - 2) \)

Use the FOIL (First, Outer, Inner, Last) method:

\( (x \cdot x) + (x \cdot -2) + (3 \cdot x) + (3 \cdot -2) \) \( = x^2 - 2x + 3x - 6 \) \( = x^2 + x - 6 \)

The result is:

\( x^2 + x - 6 \)

Practice Questions

  1. Question 1: Multiply the monomial and polynomial:

    \( 5x \cdot (3x^2 - x + 4) \)
    Solution

    Distribute \( 5x \) to each term:

    \( (5x \cdot 3x^2) + (5x \cdot -x) + (5x \cdot 4) \) \( = 15x^3 - 5x^2 + 20x \)
  2. Question 2: Multiply the binomials:

    \( (2x + 5)(x - 3) \)
    Solution

    Use the FOIL method:

    \( (2x \cdot x) + (2x \cdot -3) + (5 \cdot x) + (5 \cdot -3) \) \( = 2x^2 - 6x + 5x - 15 \) \( = 2x^2 - x - 15 \)
  3. Question 3: Multiply the binomials:

    \( (x + 4)(x + 6) \)
    Solution

    Use the FOIL method:

    \( (x \cdot x) + (x \cdot 6) + (4 \cdot x) + (4 \cdot 6) \) \( = x^2 + 6x + 4x + 24 \) \( = x^2 + 10x + 24 \)
  4. Question 4: Multiply the trinomial and the binomial:

    \( (x^2 + 3x + 2)(x - 1) \)
    Solution

    Distribute each term in the trinomial to each term in the binomial:

    \( (x^2 \cdot x) + (x^2 \cdot -1) + (3x \cdot x) + (3x \cdot -1) + (2 \cdot x) + (2 \cdot -1) \) \( = x^3 - x^2 + 3x^2 - 3x + 2x - 2 \) \( = x^3 + 2x^2 - x - 2 \)
  5. Question 5: Multiply the polynomials:

    \( (2x - 3)(x^2 + x + 4) \)
    Solution

    Distribute each term in the first polynomial to each term in the second polynomial:

    \( (2x \cdot x^2) + (2x \cdot x) + (2x \cdot 4) + (-3 \cdot x^2) + (-3 \cdot x) + (-3 \cdot 4) \) \( = 2x^3 + 2x^2 + 8x - 3x^2 - 3x - 12 \) \( = 2x^3 - x^2 + 5x - 12 \)
→Special Products of Polynomials