Factoring Basics
What You’ll Learn
Section titled “What You’ll Learn”In this lesson you’ll learn what factoring is, how to factor out the greatest common factor (GCF), and how to factor simple quadratic trinomials (a = 1).
The Concept
Section titled “The Concept”Factoring is the reverse of multiplying. You break a polynomial back into simpler factors that multiply to give the original.
Key types for basics:
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Greatest Common Factor (GCF): Factor out the largest number and lowest power of each variable common to all terms.
Example: 6x² + 9x − 12 = 3(2x² + 3x − 4)
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Factoring trinomials (a = 1): For x² + bx + c, find two numbers that multiply to c and add to b.
Example: x² + 7x + 12 = (x + 3)(x + 4) because 3 × 4 = 12 and 3 + 4 = 7
Steps for trinomials:
- Look for two numbers that multiply to the constant (c) and add to the middle coefficient (b)
- Write as (x + first)(x + second)
- If c is negative, one number is positive and one is negative
Always check by multiplying back (FOIL) to verify.
The FOIL Method (Checking Your Work)
Section titled “The FOIL Method (Checking Your Work)”FOIL stands for First, Outer, Inner, Last. It’s how you multiply two binomials. Since factoring is the reverse of multiplying, FOIL is your best friend for checking answers.
To multiply (x + 3)(x + 4):
- First: x · x = x²
- Outer: x · 4 = 4x
- Inner: 3 · x = 3x
- Last: 3 · 4 = 12
Combine: x² + 4x + 3x + 12 = x² + 7x + 12 ✓
So if you factor x² + 7x + 12 and get (x + 3)(x + 4), FOIL it back. If you land on the original, your factoring is correct.
FOIL can feel tricky at first. Keeping track of four multiplications and combining terms takes some getting used to. Stick with it. After a few practice problems, the pattern clicks and it becomes second nature.
Worked Example
Section titled “Worked Example”-
Factor 8x³ − 12x² + 4x
GCF = 4x → 4x(2x² − 3x + 1)
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Factor x² + 5x + 6
Find two numbers that multiply to 6 and add to 5: 2 and 3
x² + 5x + 6 = (x + 2)(x + 3)
Check: (x + 2)(x + 3) = x² + 3x + 2x + 6 = x² + 5x + 6 ✓
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Factor x² − 2x − 15
Find two numbers that multiply to −15 and add to −2: −5 and 3
x² − 2x − 15 = (x − 5)(x + 3)
Check: (x − 5)(x + 3) = x² + 3x − 5x − 15 = x² − 2x − 15 ✓
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Factor 3x² + 6x − 9
First, factor out the GCF: 3(x² + 2x − 3)
Now factor the trinomial: find two numbers that multiply to −3 and add to 2: 3 and −1
3(x² + 2x − 3) = 3(x + 3)(x − 1)
Check: 3(x + 3)(x − 1) = 3(x² − x + 3x − 3) = 3(x² + 2x − 3) = 3x² + 6x − 9 ✓
Real-World Application
Section titled “Real-World Application”Factoring helps simplify and solve real problems:
- Area of a garden: If the area is x² + 8x + 15 square feet, factoring gives (x + 3)(x + 5), so the dimensions are (x + 3) by (x + 5) feet.
- Projectile height: A ball’s height h = −16t² + 48t can be factored as h = −16t(t − 3), showing it hits the ground at t = 0 and t = 3 seconds.
- Profit equation: If profit = x² − 4x − 12, factoring gives (x − 6)(x + 2), so break-even points are x = 6 and x = −2 (only x = 6 makes sense in context).
Factoring turns complex expressions into useful pieces: dimensions, time points, or break-even values.