Lesson 8.9

Factoring Trinomials (Negative)

When the last number is negative, the signs battle. One plus, one minus. Who wins? The middle term decides.

Introduction

We are continuing with . But now, is negative (e.g., ). This means the signs in our parentheses must be DIFFERENT.

Past Knowledge

Lesson 8.8 (Factoring Positive). You mastered finding numbers. Now we just add a sign twist.

Today's Goal

Factor correctly using different signs .

Future Success

This is crucial for identifying zeros of functions in Pre-Calculus (Descartes' Rule of Signs).

Key Concepts

The Sign Rule

If the Last Number (C) is Negative:

1. Signs must be Opposite (one positive, one negative).

2. Instead of Adding to the middle, we are essentially Subtracting.

"Find two numbers that multiply to C and have a DIFFERENCE of B."

(The larger number takes the sign of the middle term)

Worked Examples

Example 1: Positive Middle

Basic

Factor:

1

The Puzzle

Multiply to -15.

Add to +2.

2

List Factors

  • 1 and 15 (Difference is 14)
  • 3 and 5 (Difference is 2) ← WINNER!
3

Assign Signs

We need +2. So the bigger number (5) must be positive.

and .

Answer

Example 2: Negative Middle

Intermediate

Factor:

1

The Puzzle

Multiply to -12.

Add to -4.

2

List Factors

  • 1 and 12 (Diff 11)
  • 2 and 6 (Diff 4) ← WINNER!
  • 3 and 4 (Diff 1)
3

Assign Signs

We need -4. So the bigger number (6) must be negative.

and .

Answer

Example 3: GCF First (Always)

Advanced

Factor:

1

GCF!

Pull out 3.

2

Factor Inside

Multiply to -20. Add to -1.

Factors 4 and 5 (Diff 1). Bigger number (5) needs to be negative.

Answer

Common Pitfalls

Wrong Sign on the Answer

In , using gives instead of . Always check your middle term!

Adding instead of Subtracting

Students see and try to use 2 and 3 because . But since it's , you must Subtract. .

Real-Life Applications

Profit vs Loss:

  • Profit models often look like quadratic equations.
  • Finding "break-even points" (where profit = 0) requires factoring.
  • Sometimes one break-even point is negative (meaning in the past/impossible) and one is positive (the real target). You need signs to tell them apart.

Practice Quiz

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