Multiplying Polynomials

Description: In this lesson, you'll learn how to multiply polynomials. We will work through six examples and provide practice exercises with solutions. By the end, you should be able to confidently multiply polynomials of various degrees.

Lesson Outline:

What is a Polynomial?
Multiplying Monomials by Polynomials
Multiplying Polynomials by Polynomials
Examples
Exercises
Homework
Revision

1. What is a Polynomial?

A polynomial is an algebraic expression made up of terms, where each term consists of a coefficient and variables raised to non-negative integer powers. For example, \( 4x^2 - 3x + 7 \) is a polynomial.

2. Multiplying Monomials by Polynomials

When multiplying a monomial (a single term) by a polynomial, you distribute the monomial across all terms in the polynomial. For example:

Example 1: Multiply \( 3x \) by \( 2x^2 - 4x + 5 \).

Solution:

Distribute \( 3x \) to each term: \[ 3x(2x^2 - 4x + 5) = 6x^3 - 12x^2 + 15x \]

3. Multiplying Polynomials by Polynomials

When multiplying polynomials, apply the distributive property to each term in the first polynomial and multiply it by each term in the second polynomial. For example:

Example 2: Multiply \( (x + 3) \) by \( (2x - 5) \).

Solution:

Use the distributive property: \[ (x + 3)(2x - 5) = x(2x - 5) + 3(2x - 5) = 2x^2 - 5x + 6x - 15 = 2x^2 + x - 15 \]