Algebra Foundations for Normal People by Gabriel Beguerie - HTML preview
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Chapter goals
By the end of this chapter, you will be able to:
• factor out a greatest common factor,
• factor simple trinomials,
• recognize a difference of squares,
• decide when an expression does not factor nicely over the integers,
• use expansion to check whether your factoring is correct.
Prerequisites (if this is hard, go back to Unit 0)
You need distribution from Chapter 5 and exponents from Chapter 9. Factoring reverses expansion.
Why this matters
Key idea
Factoring is the reverse of distribution.
Since distribution appears everywhere, factoring appears everywhere too. When an expression looks stuck, factoring often reveals the structure that was hiding inside it.
Real life: what it’s for
Factoring helps simplify formulas, solve equations, find where graphs cross an axis, and reduce expressions before plugging into a calculator or spreadsheet. Later, calculus uses factoring to simplify limits and difference quotients.
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Chapter 10. Factoring basics
Recipe (steps that always work)
A calm decision order is:
1. look for a greatest common factor first,
2. if the expression has two terms, check for a difference of squares, 3. if it has three terms of the form 𝑥 2 + 𝑏𝑥 + 𝑐, test a trinomial pattern, 4. expand your answer to check it.
Step 1: greatest common factor first
Key idea
If every term shares a common factor, pull that out first.
This is the factoring move students most often forget.
Worked example (no skipped steps)
Factor:
6 𝑥 + 9
The greatest common factor is 3.
Divide each term by 3:
6 𝑥 ÷ 3 = 2 𝑥,
9 ÷ 3 = 3
So:
6 𝑥 + 9 = 3(2 𝑥 + 3)
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Worked example (no skipped steps)
Factor:
12 𝑎 2 𝑏 + 18 𝑎𝑏
The coefficients 12 and 18 share a factor of 6. Both terms also contain 𝑎𝑏.
So the greatest common factor is:
6 𝑎𝑏
Now divide each term by 6 𝑎𝑏:
12 𝑎 2 𝑏 ÷ 6 𝑎𝑏 = 2 𝑎,
18 𝑎𝑏 ÷ 6 𝑎𝑏 = 3
So:
12 𝑎 2 𝑏 + 18 𝑎𝑏 = 6 𝑎𝑏(2 𝑎 + 3) Step 2: simple trinomials
Key idea
To factor
𝑥 2 + 𝑏𝑥 + 𝑐,
look for two integers that:
• multiply to 𝑐,
• add to 𝑏.
Worked example (no skipped steps)
Factor:
𝑥 2 + 7 𝑥 + 10
We need two numbers that multiply to 10 and add to 7.
Those numbers are 5 and 2.
So:
𝑥 2 + 7 𝑥 + 10 = ( 𝑥 + 5)( 𝑥 + 2) Check:
( 𝑥 + 5)( 𝑥 + 2) = 𝑥 2 + 2 𝑥 + 5 𝑥 + 10 = 𝑥 2 + 7 𝑥 + 10
Correct.
Chapter 10. Factoring basics
Worked example (no skipped steps)
Factor:
𝑥 2 − 9 𝑥 + 20
We need two numbers that multiply to 20 and add to −9.
Those numbers are −5 and −4.
So:
𝑥 2 − 9 𝑥 + 20 = ( 𝑥 − 5)( 𝑥 − 4) Worked example (no skipped steps)
Factor:
3 𝑥 2 + 12 𝑥
Do not start with the trinomial pattern. Start with the greatest common factor: 3 𝑥 2 + 12 𝑥 = 3 𝑥( 𝑥 + 4)
That is the correct factoring.
Step 3: difference of squares
Key idea
The pattern
𝑎 2 − 𝑏 2 = ( 𝑎 − 𝑏)( 𝑎 + 𝑏) appears often.
Watch the signs carefully:
• there are two terms,
• there is a minus sign between them,
• both terms are perfect squares.
Worked example (no skipped steps)
Factor:
𝑥 2 − 16
Recognize that:
16 = 42
So:
𝑥 2 − 16 = 𝑥 2 − 42 = ( 𝑥 − 4)( 𝑥 + 4)
When it does not factor nicely
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Worked example (no skipped steps)
Factor:
4 𝑥 2 − 25
Both terms are perfect squares:
4 𝑥 2 = (2 𝑥)2 ,
25 = 52
So:
4 𝑥 2 − 25 = (2 𝑥 − 5)(2 𝑥 + 5)
When it does not factor nicely
Key idea
Not every quadratic factors nicely over the integers.
That is not failure. That is information.
Worked example (no skipped steps)
Can
𝑥 2 + 2 𝑥 + 2
be factored using integers?
We need two integers that multiply to 2 and add to 2.
Try the integer pairs:
1 · 2 = 2 ,
1 + 2 = 3
(−1)(−2) = 2 ,
−1 + (−2) = −3
Neither pair adds to 2.
So this trinomial does not factor nicely over the integers.
Note
A common mistake is to force a pattern because the problem looks like a quadratic. If the check does not work, stop forcing it.
Bridge (why this matters later)
Factoring is one of the strongest bridges in the whole series.
In Math Functions for Normal People, factoring helps reveal zeros, intercepts, and structure.
In Calculus for Normal People, factoring often turns a messy expression into one that can actually be simplified and understood.
Chapter 10. Factoring basics
Common mistakes (and how to avoid them)
• forgetting to factor out the greatest common factor first,
• choosing numbers that multiply correctly but do not add correctly,
• forcing a difference-of-squares pattern when the signs do not match,
• confusing factoring with expanding,
• not checking by multiplying the factors back out.
Recap: what you’re taking away
When in doubt, use the same decision order every time:
1. common factor first,
2. then pattern check,
3. then expand to verify.
That habit is much stronger than trying to rely on memory alone.
Problems
10.1. Factor: 6 𝑥 + 9.
10.2. Factor: 8 𝑦 − 12.
10.3. Factor: 12 𝑎 2 𝑏 + 18 𝑎𝑏.
10.4. Factor: 14 𝑚 2 − 21 𝑚.
10.5. Factor: 𝑥 2 + 5 𝑥 + 6.
10.6. Factor: 𝑥 2 + 7 𝑥 + 10.
10.7. Factor: 𝑥 2 − 8 𝑥 + 15.
10.8. Factor: 𝑥 2 − 9 𝑥 + 20.
10.9. Factor: 𝑥 2 + 9 𝑥 + 20.
10.10. Factor: 𝑥 2 − 5 𝑥 − 24.
10.11. Factor: 𝑦 2 + 2 𝑦 − 15.
10.12. Factor: 𝑥 2 − 16.
10.13. Factor: 𝑥 2 − 25.
10.14. Factor: 4 𝑥 2 − 25.
10.15. Factor: 𝑎 2 − 16.
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10.16. Factor: 9 𝑚 2 − 1.
10.17. Factor: 3 𝑥 2 + 12 𝑥.
10.18. Factor: 2 𝑥 2 + 10 𝑥.
10.19. Expand to check: ( 𝑥 + 2)( 𝑥 + 3).
10.20. Expand to check: ( 𝑥 − 4)( 𝑥 + 4).
10.21. Decide whether 𝑥 2 + 2 𝑥 + 2 factors nicely over the integers.
10.22. Decide whether 𝑥 2 + 𝑥 − 6 factors nicely over the integers, and if so, factor it.
10.23. Factor: 5 𝑥 2 − 45.
10.24. Factor: 2 𝑥 2 + 8 𝑥 + 6.
Quick checklist (if you miss it, reread and try again)
• I look for a greatest common factor first.
• I can factor simple trinomials of the form 𝑥 2 + 𝑏𝑥 + 𝑐.
• I recognize a difference of squares.
• I know that some expressions do not factor nicely over the integers.
• I can expand my answer to check it.
Coordinates, slope, and straight lines