Factoring & the Factor Theorem

Factor polynomials and apply factor theorem. Math 30-1 Alberta Grade 12.

Lesson 2.3 of Polynomial Functions in Math 30-1 — Alberta curriculum lessons.

What you'll learn in this lesson

• Introduction to Factoring & The Factor Theorem. Discover what factoring is, why it's one of the most powerful tools in algebra, and master the Factor Theorem to quickly test for factors.
• Factoring Quadratics (a = 1). Learn to factor trinomials when the leading coefficient is 1. Master sign patterns and find zeros using detailed step-by-step examples.
• Factoring Quadratics (a ≠ 1). Master the AC Method for factoring trinomials with leading coefficients other than 1. Learn factoring by grouping step-by-step.
• The Integral Zero Theorem. Master a powerful tool for factoring higher-degree polynomials. Learn how to find integer zeros systematically using the Integral Zero Theorem.
• The Rational Root Theorem. Learn how to find all possible rational zeros of polynomials with any leading coefficient. Master the systematic approach to solving higher-degree equations.

Inside the lesson — a free preview

What is Factoring?

Factoring is the process of breaking down a polynomial into a product of simpler polynomials. Think of it as the reverse of multiplying polynomials together. Just like you can break down the number 12 into 3 × 4, you can break down polynomials into their "building blocks."

Why is factoring called "un-multiplying"?. Factoring takes us backwards from the expanded form to the factored form!

Why Do We Factor Polynomials?

Factoring is one of the most powerful tools in algebra because it helps us:

1. Find X-Intercepts (Zeros/Roots). When a polynomial is factored, finding where it equals zero becomes easy! If f(x) = (x + 2)(x - 3), then f(x) = 0 when x = -2 or x = 3.

2. Solve Polynomial Equations. Instead of solving x^2 + 5x + 6 = 0 (difficult!), we can factor to get (x + 2)(x + 3) = 0, then use the Zero Product Property: if a product equals zero, at least one factor must be zero.

3. Simplify Rational Expressions. Factoring allows us to cancel common factors in fractions like:

4. Graph Polynomial Functions. Factored form immediately tells us where the graph crosses the x-axis and helps us sketch the curve.

5. Solve Real-World Problems. From projectile motion to area problems, factoring helps us find practical solutions.

Worked example: Detailed Example 1: Using Factor Theorem to Test

Question. Is (x + 2) a factor of f(x) = x^3 + 5x^2 + 2x - 8?

Step 1: Understand what we're testing. We want to know if (x + 2) divides evenly into x^3 + 5x^2 + 2x - 8.

Step 2: Calculate f(-2). f(x) = x^3 + 5x^2 + 2x - 8

Calculate each term carefully:.

Now add them all:.

Step 3: Interpret the result. Since f(-2) = 0, by the Factor Theorem, (x + 2) IS a factor of f(x).

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