Dividing Polynomials

Use long division and synthetic division for polynomials. Grade 12 Math 30-1.

Lesson 2.2 of Polynomial Functions in Math 30-1 — Alberta curriculum lessons.

What you'll learn in this lesson

• Introduction to Polynomial Division. Learn what polynomial division is, why we use it, and understand the fundamental vocabulary and concepts before diving into the division methods.
• Long Division Method. Master the step-by-step process of polynomial long division with detailed examples and learn how to divide any polynomial by another.
• Synthetic Division - Monic Divisors. Learn the shortcut method of synthetic division for dividing polynomials by linear monic factors like (x - a). Faster and easier than long division!
• Synthetic Division - Non-Monic Divisors. Take synthetic division to the next level! Learn how to handle divisors with leading coefficients other than 1, like (2x + 3) or (3x - 5).
• The Remainder Theorem. Master the shortcut for finding remainders instantly! Learn how to use P(a) to check factors and solve problems without full division.

Inside the lesson — a free preview

What is Polynomial Division?

What is Polynomial Division?. Just like you can divide numbers (like 8 ÷ 2 = 4), you can also divide polynomials. Polynomial division is a fundamental skill that allows us to break down complex polynomial expressions into simpler parts.

Why Do We Divide Polynomials?

Why Divide Polynomials?. Dividing polynomials helps us:

The Division Vocabulary

Before we dive into the methods, let's understand the key terms. These are the same terms used in regular number division, but applied to polynomials.

The Division Statement

Every division problem can be written as a division statement:

Division Statement Formula. Dividend = Divisor × Quotient + Remainder

Using our number example: 8 = 4 × 2 + 0

Why Is This Formula EXTREMELY Important?.

Polynomial Division Statement

When dividing polynomials, the same rule applies:

Polynomial Division Statement. Dividend = Divisor × Quotient + Remainder

Two Methods for Dividing Polynomials

Two Methods for Dividing Polynomials. There are two main methods for dividing polynomials:

What Makes a "Good" Division Problem?

Not all polynomial divisions result in a remainder of zero. Just like 8 ÷ 3 = 2 remainder 2, polynomials can also have remainders.

Perfect Division (Remainder = 0).

Division with Remainder.

The Remainder Theorem

What is the Remainder Theorem?. The Remainder Theorem is a powerful shortcut that connects polynomial division with function evaluation.

Remainder Theorem. When a polynomial P(x) is divided by a binomial (x - a), then the remainder is P(a).

In other words: To find the remainder, just substitute a into the polynomial!

Two Important Cases.

What Does This Mean?. Let's break down what happens when you substitute a into the polynomial:

Example: When P(a) = 0. Let's say we want to check if (x - 3) is a factor of P(x) = x^2 - 5x + 6

Quick Example. Find the remainder when P(x) = x^3 + 2x^2 - 5x + 1 is divided by (x - 2)

Why This is AMAZING! 🎉. Instead of doing long division or synthetic division, you can:

Worked example: Complete Example 1: Step-by-Step Synthetic Division

Problem. Divide (x^3 + 8x^2 + 7x + 10) by (x + 1)

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