1. U1A L1 Examples
FACTORING REVIEW
EXAMPLES

2. Factor x2 + 3x – 4
Solve x2 + 3x – 4 = 0
What _____× _____ = – 4
and _____+ _____ = 3 𝟒 −𝟏
𝒙+𝟒 𝒙−𝟏 =𝟎
𝒙=−𝟒, 𝒙=𝟏 𝟒 −𝟏
(𝒙+𝟒)(𝒙−𝟏)
Graph Y1 = x2 + 3x – 4 Find x-intercepts
(−𝟒,0)
(𝟏,0)

3. Factor x2 + 3x + 2
What _____× _____ = 2
and _____+ _____ = 3
Solve 𝑥2 + 3𝑥 + 2=0 𝟐 𝟏
𝒙+𝟐 𝒙+𝟏 =𝟎 𝟐 𝟏
𝒙=−𝟐, 𝒙=−𝟏
(𝒙+𝟐)(𝒙+𝟏)
Factor 2x2 + 6x + 4
by taking out common factor 2
Solve 2𝑥2 +6𝑥 +4 = 0
2(x2 + 3x + 2) = 0
2(x2 + 3x + 2)
𝟐 𝒙+𝟐 𝒙+𝟏 =𝟎
𝟐(𝒙+𝟐)(𝒙+𝟏)
𝒙=−𝟐, 𝒙=−𝟏
Factor –3x2 – 9x – 6
by taking out common factor – 3
Solve −3𝑥2 −9𝑥 −6= 0
– 3(x2 + 3x + 2) = 0
– 3(x2 + 3x + 2)
−𝟑 𝒙+𝟐 𝒙+𝟏 =𝟎
−𝟑(𝒙+𝟐)(𝒙+𝟏)
𝒙=−𝟐, 𝒙=−𝟏

4. Graph Y1 = x2 + 3x + 2
Y2 = 2x2 + 6x + 4or Y2 = 2(x2 + 3x + 2)
Y3 = –3x2 – 9x – 6 orY3 = –3(x2 + 3x + 2)
Find x-intercepts
(−𝟐,0)
(−𝟏,0)

5. Factor –x2 – 6x – 8 by taking out common factor –1
−(𝒙+𝟒)(𝒙+𝟐)
Graph Y1 = –x2 – 6x – 8
Find x-intercepts
Solve –x2 – 6x – 8 = 0
– (x2 + 6x + 8) = 0
− 𝒙+𝟒 𝒙+𝟐 =𝟎
(−𝟒,0)
(−𝟐,0)
𝒙=−𝟒, 𝒙=−𝟐

6. Can you factor x2 + 4 ?
Can you solve x2 + 4 = 0 NO
𝒙 𝟐 =−𝟒
𝒙= −𝟒
Non-Real Answer
Graph Y1 = x2 – 4
Find x-intercepts
There are NO x - intercepts

7. Can you factor 𝑥2 − 4 ?
Can you solve 𝑥2 − 4 = 0
(𝒙−𝟐)(𝒙+𝟐)
𝒙−𝟐 𝒙+𝟐 =𝟎
Difference of Squares
𝒙=𝟐, 𝒙=−𝟐 OR
Graph Y1 = x2 – 4
𝒙 𝟐 =𝟒
Find x-intercepts
𝒙=±𝟐
Using this method it VERY easy to forget BOTH answers!!!!
(−𝟐,0)
(𝟐,0)

8. by taking out common factor 2
Factor 8x2 – 18
by taking out common factor 2
( 𝟑 𝟐 ,0)
𝟐 (𝟒𝒙 𝟐 −𝟗)
(− 𝟑 𝟐 ,0)
𝟐(𝟐𝒙−𝟑)(𝟐𝒙+𝟑)
Solve 8x2 – 18 = 0
𝟐 𝟐𝒙−𝟑 𝟐𝒙+𝟑 =𝟎
𝟐𝒙−𝟑=𝟎
𝟐𝒙+𝟑=𝟎
𝟐𝒙=𝟑
𝟐𝒙=−𝟑
𝒙= 𝟑 𝟐
𝒙=− 𝟑 𝟐
Solve 8x2 – 18 = 0
Common factor 2 is positive.
Graph opens up.
8 𝒙 𝟐 =𝟏𝟖
𝒙 𝟐 = 𝟏𝟖 𝟖 = 𝟗 𝟒
𝒙=± 𝟑 𝟐

9. by taking out common factor – 1
(𝟐−𝟑𝒙)(𝟐+𝟑𝒙)
−𝟏 (𝟗𝒙 𝟐 −𝟒)
Solve 4−9𝑥2 = 0
−𝟏(𝟑𝒙−𝟐)(𝟑𝒙+𝟐)
𝟐−𝟑𝒙 𝟐+𝟑𝒙 =𝟎
𝟐−𝟑𝒙=𝟎
𝟐+𝟑𝒙=𝟎
𝟐=𝟑𝒙
𝟐=−𝟑𝒙
𝒙= 𝟐 𝟑
𝒙=− 𝟐 𝟑
(− 𝟐 𝟑 ,0)
( 𝟐 𝟑 ,0)
Solve 4−9𝑥2 = 0
−𝟗 𝒙 𝟐 =−𝟒
𝒙 𝟐 = −𝟒 −𝟗 = 𝟒 𝟗
𝒙=± 𝟐 𝟑
Common factor – 1 is negative.
Graph opens down.

10. MORE COMMON FACTORING EXAMPLES
When dividing out common factors look for the greatest common numerical factor and the smallest exponent on the variables.
2 𝑥 5 −8 𝑥 4 +6 𝑥 3
2 𝑥 3 (𝑥 2 −4 𝑥 1 +3 𝑥 𝟎 )
2 𝑥 5 −8 𝑥 4 +6 𝒙 𝟑
2 𝑥 3 (𝑥 2 −4𝑥+3)
2 𝒙 𝟑 (𝑥 5−3 −4 𝑥 4−3 +3 𝑥 𝟑−𝟑 )
2 𝑥 3 (𝑥−3)(𝑥−1)

11. MORE COMMON FACTORING EXAMPLES
When dividing out common factors look for the greatest common numerical factor and the smallest exponent on the variables.
−3 𝑥 −1 −18 𝑥 −2 +6 𝑥 −3
−3 𝑥 −1 −18 𝑥 −2 +6 𝒙 −𝟑
−3 𝒙 −𝟑 (𝑥 −1− −3 +6 𝑥 −2− −3 −2 𝑥 −𝟑−(−𝟑) )
−3 𝒙 −𝟑 (𝑥 𝟐 +6 𝑥 𝟏 −2 𝑥 𝟎 )
−3 𝒙 −𝟑 (𝑥 2 +6𝑥−2)
This example will NOT factor further.

12. MORE COMMON FACTORING EXAMPLES
When dividing out common factors look for the greatest common numerical factor and the smallest exponent on the variables.
2 𝒙 −𝟏 (𝑥 2 −4 𝑥 1 +3 𝑥 𝟎 )
2𝑥−8+6 𝑥 −1
2 𝑥 −1 (𝑥 2 −4𝑥+3)
2 𝑥 1 −8 𝒙 𝟎 +6 𝒙 −𝟏
2 𝑥 −1 (𝑥−3)(𝑥−1)
2 𝒙 −𝟏 (𝑥 1−(−1) −4 𝑥 0−(−1) +3 𝑥 −𝟏−(−𝟏) )

13. When subtracting rational exponents use a common denominator.
12 𝑥 −24 𝑥 −36 𝑥 1 3
12 𝒙 𝟏 𝟑 ( 𝑥 𝟐 −2 𝑥 𝟏 −3 𝑥 𝟎 )
12 𝑥 −24 𝑥 −36 𝒙 𝟏 𝟑
12 𝒙 𝟏 𝟑 ( 𝑥 𝟐 −2𝑥−3)
12 𝒙 𝟏 𝟑 ( 𝑥 𝟕 𝟑 − 𝟏 𝟑 −2 𝑥 𝟒 𝟑 − 𝟏 𝟑 −3 𝑥 𝟏 𝟑 − 𝟏 𝟑 )
12 𝑥 (𝑥−3)(𝑥+1)
12 𝒙 𝟏 𝟑 ( 𝑥 𝟔 𝟑 −2 𝑥 𝟑 𝟑 −3 𝑥 𝟎 )

14. When subtracting rational exponents use a common denominator.
3 𝒙 − 𝟏 𝟐 ( 𝑥 𝟒 𝟐 −9 𝑥 𝟎 )
3 𝑥 −27 𝑥 − 1 2
3 𝑥 −27 𝒙 − 𝟏 𝟐
3 𝑥 − 1 2 ( 𝑥 2 −9)
3 𝑥 − 1 2 (𝑥−3)(𝑥+3)
3 𝒙 − 𝟏 𝟐 ( 𝑥 𝟑 𝟐 − −𝟏 𝟐 −9 𝑥 − 𝟏 𝟐 − −𝟏 𝟐 )

15. 4(x – 5)4 – 6(x – 5)3
2(x – 5)3 [2(x – 5) – 3]
4(x – 5)4 – 6(x – 5)3
2(x – 5)3 [2x – 10 – 3]
2(x – 5)3 [2(x – 5)4-3 – 3(x – 5)3-3]
2(x – 5)3 (2x – 13)
2(x – 5)3 [2(x – 5)1 – 3(x – 5)0]

16. 6 𝑥−2 −5 −24 𝑥−2 −6
6 𝒙−𝟐 −5 −24 𝒙−𝟐 −6
6 𝒙−𝟐 −6 [ 𝑥−2 −5− −6 −4 𝑥−2 −6− −6 ]
6 𝒙−𝟐 −6 [ 𝑥−2 1 −4 𝑥−2 0 ]
6 𝒙−𝟐 −6 [𝑥−2−4]
6 𝑥−2 −6 (𝑥−6)

17. 8 𝑥+6 − 1 2 −6 𝑥+6 − 3 2
8 𝒙+𝟔 − 1 2 −6 𝒙+𝟔 − 3 2
2 𝒙+𝟔 − 3 2 [4 𝒙+𝟔 − 1 2 − −3 2 −3 𝒙+𝟔 − 3 2 − −3 2 ]
2 𝒙+𝟔 − 3 2 [4 𝒙+𝟔 −3 𝒙+𝟔 0 ]
2 𝒙+𝟔 − 3 2 [4 𝒙+𝟔 1 −3 𝒙+𝟔 0 ]
2 𝒙+𝟔 − 3 2 [4𝑥+24−3]
2 𝑥+6 − 3 2 [4𝑥+21]

18. Factor by Decomposition Example
6x2 – 11x + 3
6 𝑥 2 −11𝑥+3
6 𝑥 2 +2𝑥−9𝑥+3
𝟔×𝟑=18
2×−9=18
2+(−9)=−7
2𝑥(3𝑥+1)−3(3𝑥+1)
(2𝑥−3)(3𝑥+1)

19. Quadratic Formula ax2 + bx + c = 0
𝒙= −𝒃± 𝒃 𝟐 −𝟒𝒂𝒄 𝟐𝒂

20. Solve for x 3x2 – 2x – 4 = 0 𝑥= −(−2)± (−2) 2 −4(3)(−4) 2(3)
𝑥= −(−2)± (−2) 2 −4(3)(−4) 2(3)
𝑥= 2±
𝑥= 2± 4×13 6
𝑥= 2±
𝑥= 2±
𝑥= 1±
Answers in simplest
and exact radical form
Approximate decimal answers
to nearest hundredth.
𝑥=−0.95 𝑎𝑛𝑑 𝑥=1.54

21. 𝑥= −(−3)± (−3) 2 −4(5)(10) 2(5) 𝑥= 3± 9−200 10 𝑥= 3± −191 6
 Solve for x x2 – 3x + 10 = 0
𝑥= −(−3)± (−3) 2 −4(5)(10) 2(5)
𝑥= 3± 9−
𝑥= 3± −191 6
Non-real answer.

22. SYNTHETIC DIVISION Divide x3 + 4x2 – 5x – 12 by x – 3
Method I: SUBTRACTION
– – –12
– 3 –21 –48
Quotient is x2 + 7x + 16
Remainder is 36
NOTE: x3 + 4x2 – 5x 12
(3)3 + 4(3)2 – 5(3) – 12 = 36

23. SYNTHETIC DIVISION Divide x3 + 4x2 – 5x – 12 by x – 3
Method II: ADDITION
– –12
Quotient is x2 + 7x + 16
Remainder is 36
NOTE: x3 + 4x2 – 5x 12
(3)3 + 4(3)2 – 5(3) – 12 = 36

24. Divide x3 + 3x2 – 5 by x + 2 SYNTHETIC DIVISION 𝒙𝟑 + 𝟑𝒙𝟐 + 𝟎𝒙 −𝟓
Method I: SUBTRACTION –5 –4
– –1
Quotient is x2 + x – 2
Remainder is –1
NOTE: x3 + 3x2 – 5
(–2)3 + 3(–2)2 – 5 = –1

25. Divide x3 + 3x2 – 5 by x + 2 SYNTHETIC DIVISION Method II: ADDITION–5
–2 –
– –1
Quotient is x2 + x – 2
Remainder is –1
NOTE: x3 + 3x2 – 5
(–2)3 + 3(–2)2 – 5 = –1

26. Divide x3 – 8 by x – 2 SYNTHETIC DIVISION 𝒙𝟑 +𝟎 𝒙 𝟐 +𝟎𝒙−𝟖
Method I: SUBTRACTION
– –8
–2 –4 –8
Quotient is x2 + 2x + 4
Remainder is 0
NOTE: x3– 8
(2)3 – 8 = 0

27. Difference of Cubes Formula
a3 – b3 = (a – b)(a2 + ab + b2)
Factor
x3 – 8
𝑎=𝑥
𝑏=2
If we compare this answer to the previous slide we see it is the same.
This is a shortcut that will help with
more difficult questions.
(𝑥−2)( 𝑥 2 +2𝑥+ 2 2 )
(𝑥−2)( 𝑥 2 +2𝑥+4)
Factor
27x3 – 64
𝑎=3𝑥
𝑏=4
3𝑥−4 [ 3𝑥 2 +(3𝑥)(4)+ 4 2 ]
3𝑥−4 (9 𝑥 2 +12𝑥+16)

28. Divide x3 + 27 by x + 3 SYNTHETIC DIVISION Method I: SUBTRACTION
3 –
1 –
Quotient is x2 – 3x + 9
Remainder is 0
NOTE: x3+ 27
(–3) = 0

29. Sum of Cubes Formula a3 + b3 = (a + b)(a2 – ab + b2)
Factor
x3 + 27
𝑎=𝑥
𝑏=3
If we compare this answer to the previous slide we see it is the same.
This is a shortcut that will help with
more difficult questions.
(𝑥+3 )( 𝑥 2 −3𝑥+ 3 2 )
(𝑥+3)( 𝑥 2 −3𝑥+9)
Factor
𝑥3+125 𝑦 3
Factor
27𝑥3−64
𝑎=𝑥
𝑏=5𝑦
𝑎=3𝑥
𝑏=4
3𝑥−4 [ 3𝑥 2 +(3𝑥)(4)+ (4) 2 ]
𝑥+5𝑦 [ 𝑥 2 −(𝑥)(5𝑦)+ (5𝑦) 2 ]
3𝑥−4 (9 𝑥 2 +12𝑥+16)
𝑥+5𝑦 ( 𝑥 2 −5𝑥𝑦+25 𝑦 2 )