Lesson 1.8

Factoring Review

Solving quadratics often means "un-multiplying" them back into their original pieces. We review the essential skills of GCF and factoring trinomials.

Introduction

Factoring is the reverse of distribution (FOIL). Just as division "undoes" multiplication for numbers, factoring "undoes" multiplication for polynomials. It is the primary tool we use to solve standard form equations.

Past Knowledge

You know how to multiply to get .

Today's Goal

We learn to go backward: Start with and break it into .

Future Success

Factoring is the fastest way to find roots (x-intercepts) and solve vertical motion problems without graphing.

Key Concepts

1. Greatest Common Factor (GCF)

Always check for a GCF first. It's the largest number or variable that divides evenly into every term.

Example:

Both terms divide by .

2. Factoring Trinomials (a=1)

For , we look for two numbers that:

  • 1Multiply to (the constant)
  • 2Add to (the middle coefficient)

3. Difference of Squares

A special pattern where two perfect squares are subtracted. There is no middle term because it cancels out.

Example

4. Factoring by Grouping

For , multiply . Find factors that add to .

The "ac" Method:

  1. Multiply .
  2. Split the middle term .
  3. Factor by grouping.

Worked Examples

Example 1: GCF Factoring

Basic

Factor .

1

Identify GCF

Coefficients: 4 and 12. GCF is 4.
Variables: and . GCF is .
Total GCF = .

2

Divide Each Term

Result:

Example 2: Trinomial Factoring

Intermediate

Factor .

1

Multiply to 12

Factors of 12: (1, 12), (2, 6), (3, 4).
Since middle term is negative, both must be negative: (-1, -12), (-2, -6), (-3, -4).

2

Add to -7

-1 + -12 = -13 (No)
-2 + -6 = -8 (No)
-3 + -4 = -7 (Yes)

Result:

Example 3: Factoring by Grouping ()

Advanced

Factor .

1

Multiply a ยท c

. Product is .
Find factors of 6 that add to 7 (the middle term).
6 and 1 work (, ).

2

Split & Group

Rewrite as :

Group terms:

3

Factor Out GCFs

Now pull out the common binomial .

Result:

Common Pitfalls

Sum of Squares doesn't factor

factors to .
is PRIME. It cannot be factored with real numbers.

Missing the GCF

Sometimes students try to reverse-foil a problem like immediately. Always pull the GCF first: . It makes the rest of the problem much easier.

Real-Life Applications

Area Reconstruction

If a landscape architect knows the total area of a garden is square meters, simpler dimensions are needed to actually build it.

Factoring reveals the dimensions are meters by meters, allowing them to scale the project for any value of .

Practice Quiz

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