Factoring Trinomials: x² + bx + c

Factor simple trinomials using reverse FOIL. Math 10-C Alberta mathematics.

Lesson 4.4 of Factors and Roots in Math 10C — Alberta curriculum lessons.

What you'll learn in this lesson

• The Connection Between Expanding and Factoring. See how factoring is simply expanding in reverse, and learn the core rule for factoring any trinomial of the form x squared plus bx plus c.
• Factoring When All Signs Are Positive or Mixed. Work through five fully solved examples covering positive constants, negative constants, and mixed signs, plus a sign guide for handling every case.
• Special Situations. Learn how to rearrange non-standard trinomials, factor out the GCF before factoring, and extend the method to trinomials with two variables.

Inside the lesson — a free preview

Factoring Is Expanding in Reverse

The Core Idea. When you expand (x + 3)(x + 1), you get x^2 + 4x + 3.

Factoring asks the opposite question: given x^2 + 4x + 3, what two binomials multiply to give it?

The answer is (x + 3)(x + 1) — and the key is noticing how the numbers in the binomials relate to the trinomial.

How the Numbers Connect. Look at (x + 3)(x + 1) = x^2 + 4x + 3:

The constant term 3 is the product of 3 and 1: 3 × 1 = 3

The middle coefficient 4 is the sum of 3 and 1: 3 + 1 = 4

This pattern always holds for trinomials of the form x^2 + bx + c.

The Rule

Factoring Rule for x^2 + bx + c. To factor x^2 + bx + c, find two numbers p and q such that:

Multiply to c (the constant term) and add to b (the middle coefficient)

If those two numbers are p and q, then:

x^2 + bx + c = (x + p)(x + q)

Why This Works. Expanding (x + p)(x + q) using FOIL gives:

x^2 + qx + px + pq

= x^2 + (p + q)x + pq

So the middle coefficient b = p + q and the constant c = p × q.

Factoring simply runs this process backwards.

Worked example: Worked Examples

Example 1 — Factor x^2 + 6x + 8. Step 1: Find two numbers that multiply to 8 and add to 6.

Step 2: List factor pairs of 8: 1 × 8, 2 × 4

Step 3: Which pair adds to 6? 2 + 4 = 6 ✓

Step 4: Write the factors:

Answer: x^2 + 6x + 8 = (x + 2)(x + 4)

Verify: x^2 + 4x + 2x + 8 = x^2 + 6x + 8 ✓

Example 2 — Factor x^2 + 12x + 20. Step 1: Find two numbers that multiply to 20 and add to 12.

Step 2: List factor pairs of 20: 1 × 20, 2 × 10, 4 × 5

Step 3: Which pair adds to 12? 2 + 10 = 12 ✓

Answer: x^2 + 12x + 20 = (x + 2)(x + 10)

Example 3 — Factor x^2 - 2x - 8. Here c is negative (-8), which means the two numbers have opposite signs. Since b = -2, the negative number has the larger absolute value.

Step 1: List factor pairs of -8: (-1)(8), (1)(-8), (-2)(4), (2)(-4)

Step 2: Which pair adds to -2? -4 + 2 = -2 ✓

Answer: x^2 - 2x - 8 = (x - 4)(x + 2)

Verify: x^2 + 2x - 4x - 8 = x^2 - 2x - 8 ✓

Example 4 — Factor a^2 + 7a - 18. c = -18 (opposite signs required); b = +7 (positive number has larger absolute value).

Step 1: List factor pairs of -18:

(-1)(18),\ (1)(-18),\ (-2)(9),\ (2)(-9),\ (-3)(6), (3)(-6)

Step 2: Which pair adds to 7? 9 + (-2) = 7 ✓

Answer: a^2 + 7a - 18 = (a + 9)(a - 2)

Example 5 — Factor x^2 - 29x + 28. c = +28 (same signs); b = -29 (both numbers must be negative — they add to a negative and multiply to a positive).

Step 1: List negative factor pairs of 28: (-1)(-28), (-2)(-14), (-4)(-7)

Step 2: Which pair adds to -29? -1 + (-28) = -29 ✓

Answer: x^2 - 29x + 28 = (x - 1)(x - 28)

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