Supp. Math

FACTORIZATION OF POLYNOMIALS — COMPLETE SUMMARY

CONTENTS

1. What is Factorization?

2. Greatest Common Factor (GCF)

3. Factoring by Grouping

4. Difference of Two Squares

5. Perfect-Square Trinomials

6. Factoring x^2+bx+c

7. Factoring ax^2+bx+c

8. AC Method / Splitting the Middle Term

9. Factoring by Substitution

10. Sum and Difference of Cubes

11. Combining Factorization Methods

12. Factoring Completely

13. Solving Equations by Factoring

14. Common Mistakes

15. How to Choose the Correct Method

16. Mixed Exercises

17. Quiz 1

18. Quiz 2

19. Challenge Questions

20. Final Cheat Sheet

1. WHAT IS FACTORIZATION?
Factorization means changing an expression from addition/subtraction into multiplication.
Example:
$x^2+5x+6$
becomes
$(x+2)(x+3)$
because:
$(x+2)(x+3)=x^2+5x+6$

Important vocabulary

• Factor = something being multiplied

• Polynomial = expression containing terms with variables and powers

• Term = parts separated by + or −

• Coefficient = number in front of a variable

• Constant = number without a variable

• Degree = highest exponent

BIG RULE
Always check for a GCF first.

2. GREATEST COMMON FACTOR (GCF)
The GCF is the biggest factor that every term has in common.
Example
$12x^3+18x^2$
The GCF is:
$6x^2$
So:
$12x^3+18x^2=6x^2(2x+3)$

Steps

1. Find the GCF of the numbers.

2. Find the variables common to every term.

3. Put the GCF outside brackets.

4. Divide each term by the GCF.

Example
$15x^2+25x$
GCF = 5x
$\boxed{5x(3x+5)}$

Practice
Factor:

1. 8x+20

2. 18x^3-24x^2

3. 14x^2+21x

Answers

1. 4(2x+5)

2. 6x^2(3x-4)

3. 7x(2x+3)

3. FACTORING BY GROUPING
Usually used when there are 4 terms.
Steps

1. Group the first two terms.

2. Group the last two terms.

3. Factor each group.

4. Find the common bracket.

5. Factor again.

Example
$x^3+3x^2+2x+6$
Group:
$(x^3+3x^2)+(2x+6)$
Factor:
$x^2(x+3)+2(x+3)$
Common factor:
$\boxed{(x+3)(x^2+2)}$

Another example
$6x^2+9x+4x+6$
$3x(2x+3)+2(2x+3)$
$\boxed{(3x+2)(2x+3)}$

Practice
Factor:

1. $x^3+2x^2+5x+10$

2. $3x^2-6x+2x-4$

3. $ax+ay+bx+by$

Answers

1. $(x+2)(x^2+5)$

2. $(3x+2)(x-2)$

3. $(a+b)(x+y)$

4. DIFFERENCE OF TWO SQUARES
Formula
$\boxed{a^2-b^2=(a-b)(a+b)}$
It has two requirements:

• There are 2 terms.

• They are being subtracted.

• Both terms are perfect squares.

Examples
$x^2-25$
= $x^2-5^2$
$\boxed{(x-5)(x+5)}$
Another:
$9x^2-16$
= $(3x)^2-4^2$
$\boxed{(3x-4)(3x+4)}$

DON’T use it for:
$x^2+25$
because it is a sum, not a difference.

Practice

1. $x^2-49$

2. $4x^2-81$

3. $16a^4-25b^2$

Answers

1. $(x-7)(x+7)$

2. $(2x-9)(2x+9)$

3. $(4a^2-5b)(4a^2+5b)$

5. PERFECT-SQUARE TRINOMIALS
There are two important formulas.
Formula 1
$\boxed{a^2+2ab+b^2=(a+b)^2}$
Formula 2
$\boxed{a^2-2ab+b^2=(a-b)^2}$
How to recognize one
Check:

1. Is the first term a square?

2. Is the last term a square?

3. Is the middle term 2ab or -2ab?

Example
$x^2+10x+25$
Square roots:
$ext{ } \sqrt{x^2}=x <br>\sqrt{25}=5$
Middle:
$2(x)(5)=10x$
Therefore:
$\boxed{(x+5)^2}$
Example
$4x^2-12x+9$
ext{ } $=(2x)^2-2(2x)(3)+3^2$
$\boxed{(2x-3)^2}$
Sign trick
Positive middle:
$(a+b)^2$
Negative middle:
$(a-b)^2$

6. FACTORING x^2+bx+c
This is one of the MOST IMPORTANT methods.
For:
$x^2+bx+c$
find two numbers that:
ADD to b
and
MULTIPLY to c
Example
$x^2+7x+12$
We need:

• Add = 7

• Multiply = 12
  Numbers:
  3, 4
  Therefore:
  $\boxed{(x+3)(x+4)}$
  Example with negatives
  $x^2-x-12$
  Need numbers that:

• Add to -1

• Multiply to -12
  Numbers:
  -4, 3
  Therefore:
  $\boxed{(x-4)(x+3)}$
  SIGN TRICK
  If c is positive:
  → same signs.
  If c is negative:
  → opposite signs.

Practice
Factor:

1. $x^2+9x+20$

2. $x^2-8x+15$

3. $x^2+2x-15$

4. $x^2-11x+24$

Answers

1. $(x+4)(x+5)$

2. $(x-3)(x-5)$

3. $(x+5)(x-3)$

4. $(x-3)(x-8)$

7. FACTORING ax^2+bx+c
Now the coefficient of x^2 isn’t 1.
Example:
$6x^2+11x+3$
This is harder than $x^2+bx+c$.
The AC method is useful.

8. AC METHOD / SPLITTING THE MIDDLE TERM
For:
$ax^2+bx+c$
Steps
1. Multiply a×c.**
2. Find two numbers that multiply to ac.**
3. Those numbers must add to b.**
4. Split the middle term.
5. Factor by grouping.
Example
$6x^2+11x+3$
First:
$6 imes3=18$
Find numbers that multiply to 18 and add to 11:
9, 2
Split:
$6x^2+9x+2x+3$
Group:
$3x(2x+3)+1(2x+3)$
Therefore:
$\boxed{(3x+1)(2x+3)}$
Example
$2x^2+7x+3$
$2 imes3=6$
Numbers:
6, 1
Split:
$2x^2+6x+x+3$
Group:
$2x(x+3)+1(x+3)$
Answer:
$\boxed{(2x+1)(x+3)}$

9. FACTORING BY SUBSTITUTION
This is useful when you see repeated powers.
Example:
$x^4+5x^2+6$
Notice:
$x^4=(x^2)^2$
Let:
u$=x^2$
Then:
$u^2+5u+6$
Factor:
$(u+2)(u+3)$
Replace u:
$\boxed{(x^2+2)(x^2+3)}$
Another example
$x^6-9x^3+20$
Let:
u$=x^3$
Then:
$u^2-9u+20$
$(u-4)(u-5)$
Therefore:
$\boxed{(x^3-4)(x^3-5)}$
Trick
If you see:
$x^4, x^2$
think: Let u=x^2.
If you see:
$x^6, x^3$
think: Let u=x^3.

10. SUM AND DIFFERENCE OF CUBES
Sum of cubes
$\boxed{a^3+b^3=(a+b)(a^2-ab+b^2)}$
Difference of cubes
$\boxed{a^3-b^3=(a-b)(a^2+ab+b^2)}$
Example
$x^3+8$
Since:
8=2^3
$x^3+2^3$
Therefore:
$\boxed{(x+2)(x^2-2x+4)}$
Example
$x^3-27$
= $x^3-3^3$
$\boxed{(x-3)(x^2+3x+9)}$
SOAP trick
Same
Opposite
Always
Positive
For cubes:
First bracket → same sign.
Second bracket → opposite sign.
Last term → always positive.

11. COMBINING METHODS
Some questions need more than one method.
Example
$2x^3-18x$
First take GCF:
$2x(x^2-9)$
Now difference of squares:
$x^2-9=(x-3)(x+3)$
Final answer:
$\boxed{2x(x-3)(x+3)}$
Example
$x^3+3x^2-4x-12$
Group:
$x^2(x+3)-4(x+3)$
ext{''} = $(x+3)(x^2-4)$
Difference of squares:
$\boxed{(x+3)(x-2)(x+2)}$
Important
After factoring once, look at every factor again.

12. FACTORING COMPLETELY
A polynomial is completely factored when nothing else can be factored.
Example
$12x^4-75x^2$
GCF:
$3x^2(4x^2-25)$
Difference of squares:
$3x^2(2x-5)(2x+5)$
Final:
$\boxed{3x^2(2x-5)(2x+5)}$
Don’t stop here:
$3x^2(4x^2-25)$
because $4x^2-25$ can still be factored.

13. SOLVING EQUATIONS BY FACTORING
Factoring can help solve quadratic equations.
Zero Product Property
If:
$AB=0$
then:
$A=0$
or:
$B=0$
Example
$x^2+5x+6=0$
Factor:
$(x+2)(x+3)=0$
Therefore:
$x+2=0$
or:
$x+3=0$
So:
$\boxed{x=-2, -3}$
Example
$2x^2-8x=0$
GCF:
$2x(x-4)=0$
Therefore:
$2x=0$
or:
$x-4=0$
Answers:
$\boxed{x=0, 4}$
Important warning
Don’t divide by x immediately.
For example:
$x^2-4x=0$
If you divide by x, you might lose the solution $x=0$.

14. COMMON MISTAKES
Mistake 1: Forgetting GCF
Wrong:
$6x^2+12x=x(6x+12)$
Better:
$\boxed{6x(x+2)}$
Mistake 2: Wrong signs
For:
$x^2-5x+6$
Correct:
$\boxed{(x-2)(x-3)}$
because:
-2+-3=-5
and:
(-2)(-3)=6
Mistake 3: Using difference of squares on a sum
Wrong:
$x^2+25=(x+5)(x-5)$
because:
$(x+5)(x-5)=x^2-25$
Mistake 4: Stopping too early
$x^4-16$
First:
$(x^2-4)(x^2+4)$
Then:
$\boxed{(x-2)(x+2)(x^2+4)}$
Mistake 5: Wrong cube signs
Remember:
$a^3-b^3=(a-b)(a^2+ab+b^2)$
The second bracket has plus signs.

15. HOW TO CHOOSE THE METHOD
QUICK DECISION GUIDE
2 terms?
Check:
GCF → difference/sum of squares → cubes
3 terms?
Check:
GCF → perfect square → trinomial
4 terms?
Try:
GCF → grouping
x^4 and x^2?
Try:
substitution
x^6 and x^3?
Try:
substitution
ax^2+bx+c?
Try:
AC method

THE BEST EXAM ROUTINE
Whenever you see a factorization question:
STEP 1
GCF?
STEP 2
How many terms?
STEP 3
Look for a special pattern.
STEP 4
Factor.
STEP 5
Factor again if possible.
STEP 6
Multiply back to check.

16. MIXED EXERCISES
Factor completely.

1. $14x^2+21x$

2. $x^2-36$

3. $x^2+13x+40$

4. $x^2-2x-24$

5. $2x^2+9x+4$

6. $6x^2-x-2$

7. $x^3+27$

8. $x^3-8$

9. $x^4-81$

10. $4x^3-36x$

11. $x^3+4x^2-9x-36$

12. $3x^4-48x^2$

13. $x^4-5x^2+4$

14. $6x^3+7x^2-6x-7$

15. $12x^2+11x-5$

Answers

1. $7x(2x+3)$

2. $(x-6)(x+6)$

3. $(x+5)(x+8)$

4. $(x-6)(x+4)$

5. $(2x+1)(x+4)$

6. $(3x-2)(2x+1)$

7. $(x+3)(x^2-3x+9)$

8. $(x-2)(x^2+2x+4)$

9. $(x-3)(x+3)(x^2+9)$

10. $4x(x-3)(x+3)$

11. $(x+4)(x-3)(x+3)$

12. $3x^2(2x-4)(2x+4) o ext{further: } \boxed{12x^2(x-2)(x+2)}$

13. $(x^2-1)(x^2-4)=(x-1)(x+1)(x-2)(x+2)$

14. $(3x+7)(2x^2-1)$

15. $(3x-1)(4x+5)$

17. QUIZ 1 — BASICS
Try without looking at the notes.
1.
What should you usually check first?
2.
Factor:
$8x+20$
3.
Factor:
$x^2-64$
4.
Factor:
$x^2+12x+36$
5.
Factor:
$x^2+5x+6$
6.
Factor:
$x^2-9x+20$
7.
Can $x^2+16$ be factored using the difference-of-squares formula?
8.
Factor:
$x^3-125$
9.
Factor:
$2x^2+7x+3$
10.
What property says that if $AB=0$, then $A=0$ or $B=0$?
Answers

1. GCF

2. $4(2x+5)$

3. $(x-8)(x+8)$

4. $(x+6)^2$

5. $(x+2)(x+3)$

6. $(x-4)(x-5)$

7. No

8. $(x-5)(x^2+5x+25)$

9. $(2x+1)(x+3)$

10. Zero Product Property

18. QUIZ 2 — MIXED
1.
$18x^3-50x$
2.
$9x^4-16$
3.
$4x^2+4x-15$
4.
$6x^2-17x+12$
5.
$x^4+2x^2-15$
6.
$x^3+6x^2+9x+54$
7. Solve
$x^2-10x+21=0$
8. Solve
$2x^2-18=0$
9.
Why can’t $x^2+9$ be factored into real linear binomials?
10.
Correct this:
$x^2-16=(x-4)^2$
Answers

1. $2x(3x-5)(3x+5)$

2. $(3x-2)(3x+2)(3x^2+4)$

3. $(2x+5)(2x-3)$

4. $(3x-4)(2x-3)$

5. $(x^2+5)(x^2-3)$

6. $(x+6)(x^2+9)$

7. $x=3,7$

8. $x=-3,3$

9. It is a sum of squares and has no real linear factors.

10. $\boxed{(x-4)(x+4)}$

19. CHALLENGE QUESTIONS
1.
Factor completely:
$2x^4-18x^2$
2.
Factor completely:
$3x^3-12x^2+12x$
3.
Factor completely:
$x^6-64$
4.
Factor:
$4x^4-20x^2+25$
5.
Factor:
$6x^3+7x^2-6x-7$
6.
Solve:
$x^3-4x^2-9x+36=0$
Answers

1. $\boxed{2x^2(x-3)(x+3)}$

2. $\boxed{3x(x-2)^2}$

3. $\boxed{(x-2)(x+2)(x^2+2x+4)(x^2-2x+4)}$

4. $\boxed{(2x^2-5)^2}$

5. $\boxed{(3x+7)(2x^2-1)}$

6. $\boxed{x=4,3,-3}$

20. FINAL CHEAT SHEET
MUST KNOW
GCF
$ab+ac=a(b+c)$
Difference of squares
$\boxed{a^2-b^2=(a-b)(a+b)}$
Perfect square
$\boxed{a^2+2ab+b^2=(a+b)^2}$
$\boxed{a^2-2ab+b^2=(a-b)^2}$
Sum of cubes
$\boxed{a^3+b^3=(a+b)(a^2-ab+b^2)}$
Difference of cubes
$\boxed{a^3-b^3=(a-b)(a^2+ab+b^2)}$
x^2+bx+c
Find numbers that:
$\boxed{ ext{ADD}=b} <br>$oxed{ ext{MULTIPLY}=c}$<br><strong>ax^2+bx+c</strong> <br>Find numbers that: <br>$oxed{ ext{ADD}=b}
$\boxed{ ext{MULTIPLY}=ac}$
Then split the middle term and group.
⭐ TOP 10 TRICKS TO REMEMBER

1. Always check GCF first.

2. 2 terms → check special formulas.

3. 3 terms → think trinomial.

4. 4 terms → think grouping.

5. $a^2-b^2 → ext{difference of squares}$.

6. $a^2+2ab+b^2 → ext{perfect square}$.

7. Cubes → remember SOAP.

8. Negative product → numbers have opposite signs.

9. Positive product → numbers have same signs.

10. Always factor again after your first step.

The most important habit:
GCF → Pattern → Factor → Factor again → Check
If you can follow that order, factorization becomes much easier.

Formulas for Factorization

1. Greatest Common Factor (GCF)

   • $ab + ac = a(b + c)$

2. Difference of Squares

   • $a^2 - b^2 = (a - b)(a + b)$

3. Perfect Square Trinomials

   • $a^2 + 2ab + b^2 = (a + b)^2$

   • $a^2 - 2ab + b^2 = (a - b)^2$

4. Sum of Cubes

   • $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$

5. Difference of Cubes

   • $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$

6. Factoring Trinomials

   • For $x^2 + bx + c$: Find numbers that

     • ADD = $b$

     • MULTIPLY = $c$

   • For $ax^2 + bx + c$: Find numbers that

     • ADD = $b$

     • MULTIPLY = $ac$

     • Then split the middle term and group.

7. Zero Product Property

   • If $AB = 0$, then $A = 0$ or $B = 0$

8. Special Case for Factoring:

   • For repeated powers, if $x^n + k$ can be factored by substitution where $u = x^{n/2}$ (for even n) or $u = x^{n/3}$ (for multiples of 3).