Factoring Polynomials: GCF and Trinomials

Factor out common terms from polynomials. Math 10-C Alberta mathematics lesson.

Lesson 4.3 of Factors and Roots in Math 10C — Alberta curriculum lessons.

What you'll learn in this lesson

• Factoring and Expanding Are Opposites. Discover why factoring and expanding are inverse operations, and how to think about polynomials as rectangular areas made of tiles.
• Factoring Out the GCF. Learn the five-step method for removing the greatest common factor from any polynomial, with five fully worked examples including binomials, monomials, and trinomials.
• Factoring with Multiple Variables and Negative Leading Terms. Extend GCF factoring to polynomials with two variables and practice the convention of factoring out a negative GCF so the leading term inside the brackets is positive.
• Introduction to Factoring Trinomials. See how some polynomials factor into a product of two binomials, and learn the key question: which two numbers multiply to c and add to b?

Inside the lesson — a free preview

The Big Idea

Factoring vs. Expanding. Factoring and expanding are inverse operations — they undo each other.

When you expand, you distribute to remove brackets.

When you factor, you find what is common and pull it out to create brackets.

If you factor correctly, expanding your answer should give you back the original expression — this is how you verify your work.

Key Takeaway. Expanding goes left to right: 3(2 - 5a) → 6 - 15a

Factoring goes right to left: 6 - 15a → 3(2 - 5a)

How Factoring Works Visually

The Tile Model. Think of a polynomial like tiles that can be arranged into a rectangle.

The length and width of the rectangle are the factors, and the area is the polynomial.

For the polynomial 8x + 4, here are three ways to arrange the tiles:

Three Ways to Factor 8x + 4. 1 × (8x + 4) — valid, but not simplified

2 × (4x + 2) — partly factored

4 × (2x + 1) — fully factored (no more common factors inside the brackets)

What Does Fully Factored Mean?. Fully factored means the GCF has been completely removed.

The fully factored form of 8x + 4 is 4(2x + 1) because 2x + 1 cannot be factored further.

If the terms inside the brackets still share a common factor, you have not finished.

Worked example: Worked Examples

Example 1 — Factor 3x + 6. Step 1: GCF of 3 and 6 is 3. No variable is common to both terms.

Step 2: GCF = 3

Step 3: Divide: 3x ÷ 3 = x, 6 ÷ 3 = 2

Step 4: 3x + 6 = 3(x + 2)

Step 5 (Verify): 3(x) + 3(2) = 3x + 6 ✓

Example 2 — Factor 16x - 4. Step 1: GCF of 16 and 4 is 4.

Step 2: GCF = 4

Step 3: Divide: 16x ÷ 4 = 4x, -4 ÷ 4 = -1

Step 4: 16x - 4 = 4(4x - 1)

Step 5 (Verify): 4(4x) + 4(-1) = 16x - 4 ✓

Example 3 — Factor 8d + 12d^2. Step 1: GCF of 8 and 12 is 4.

Step 2: Both terms have d — lowest power is d^1. GCF = 4d.

Step 3: Divide: 8d ÷ 4d = 2, 12d^2 ÷ 4d = 3d

Step 4: 8d + 12d^2 = 4d(2 + 3d)

Step 5 (Verify): 4d(2) + 4d(3d) = 8d + 12d^2 ✓

Example 4 — Factor 9x^4 - 27x^2. Step 1: GCF of 9 and 27 is 9.

Step 2: Both terms have x — lowest power is x^2. GCF = 9x^2.

Step 3: Divide: 9x^4 ÷ 9x^2 = x^2, -27x^2 ÷ 9x^2 = -3

Step 4: 9x^4 - 27x^2 = 9x^2(x^2 - 3)

Step 5 (Verify): 9x^2(x^2) + 9x^2(-3) = 9x^4 - 27x^2 ✓

Example 5 — Factor the Trinomial 6 - 12z + 18z^2. Step 1: GCF of 6, 12, and 18 is 6.

Step 2: The first term (6) has no z, so z is not in the GCF. GCF = 6.

Step 3: Divide: 6 ÷ 6 = 1, -12z ÷ 6 = -2z, 18z^2 ÷ 6 = 3z^2

Step 4: 6 - 12z + 18z^2 = 6(1 - 2z + 3z^2)

Note: The 1 inside the brackets comes from 6 ÷ 6. Never drop it!

Step 5 (Verify): 6(1) + 6(-2z) + 6(3z^2) = 6 - 12z + 18z^2 ✓

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