factoring review examples 1. factor x 2 + 3x – 4solve x 2 + 3x – 4 = 0 graph y 1 = x 2 + 3x –...
TRANSCRIPT
1
FACTORING REVIEW
EXAMPLES
2
Factor x2 + 3x – 4 Solve x2 + 3x – 4 = 0
Graph Y1 = x2 + 3x – 4 Find x-intercepts
What _____× _____ = – 4 and _____+ _____ = 3−𝟏𝟒
−𝟏𝟒
(𝒙+𝟒)(𝒙 −𝟏)
(−𝟒 ,0) (𝟏 ,0)
3
Factor x2 + 3x + 2 What _____× _____ = 2 and _____+ _____ = 3
Factor 2x2 + 6x + 4 by taking out common factor 2
Factor –3x2 – 9x – 6 by taking out common factor – 3
Solve
𝟏𝟐𝟏𝟐
(𝒙+𝟐)(𝒙+𝟏)
𝟐(𝒙+𝟐)(𝒙+𝟏)2(x2 + 3x + 2)
– 3(x2 + 3x + 2)
−𝟑 (𝒙+𝟐)(𝒙+𝟏)
𝒙=−𝟐 ,𝒙=−𝟏( 𝒙+𝟐 ) ( 𝒙+𝟏 )=𝟎
Solve
2(x2 + 3x + 2) = 0
𝟐 ( 𝒙+𝟐 ) (𝒙+𝟏 )=𝟎𝒙=−𝟐 ,𝒙=−𝟏
– 3(x2 + 3x + 2) = 0
Solve
−𝟑 ( 𝒙+𝟐 ) ( 𝒙+𝟏 )=𝟎
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)
(−𝟐 ,0) (−𝟏 ,0)Find x-intercepts
5
Factor –x2 – 6x – 8 by taking out common factor –1
Solve –x2 – 6x – 8 = 0
Graph Y1 = –x2 – 6x – 8
– (x2 + 6x + 8)
−(𝒙+𝟒)(𝒙+𝟐)
– (x2 + 6x + 8) = 0
− ( 𝒙+𝟒 ) ( 𝒙+𝟐 )=𝟎𝒙=−𝟒 , 𝒙=−𝟐
Find x-intercepts
(−𝟒 ,0) (−𝟐 ,0)
Can you factor x2 + 4 ? Can you solve x2 + 4 = 0
Graph Y1 = x2 – 4
6
NO 𝒙𝟐=−𝟒𝒙=√−𝟒
Non-Real Answer
Find x-intercepts
There are NO x - intercepts
Can you factor ? Can you solve
Graph Y1 = x2 – 4
7
(𝒙 −𝟐)(𝒙+𝟐)
𝒙𝟐=𝟒
𝒙=±𝟐Find x-intercepts
Difference of Squares
( 𝒙 −𝟐 ) (𝒙+𝟐 )=𝟎𝒙=𝟐 , 𝒙=−𝟐
(−𝟐 ,0) (𝟐 ,0)
Using this method it VERY easy to forget BOTH answers!!!!
OR
8
Factor 8x2 – 18 by taking out common factor 2
Solve 8x2 – 18 = 0
𝟐(𝟒 𝒙¿¿𝟐−𝟗)¿𝟐(𝟐 𝒙 −𝟑)(𝟐 𝒙+𝟑)
8
𝒙𝟐=𝟏𝟖𝟖
=𝟗𝟒
𝒙=±𝟑𝟐
Solve 8x2 – 18 = 0
𝟐 (𝟐 𝒙 −𝟑 ) (𝟐 𝒙+𝟑 )=𝟎𝟐 𝒙 −𝟑=𝟎 𝟐 𝒙+𝟑=𝟎𝟐 𝒙=𝟑 𝟐 𝒙=−𝟑𝒙=
𝟑𝟐
𝒙=−𝟑𝟐
Common factor 2 is positive.Graph opens up.
(𝟑𝟐
, 0)(−𝟑𝟐
,0)
9
Factor
(𝟐−𝟑 𝒙)(𝟐+𝟑 𝒙)
𝒙𝟐=−𝟒−𝟗
=𝟒𝟗
𝒙=±𝟐𝟑
Solve = 0
(𝟐−𝟑 𝒙 ) (𝟐+𝟑 𝒙 )=𝟎
𝟐−𝟑 𝒙=𝟎 𝟐+𝟑 𝒙=𝟎𝟐=𝟑 𝒙 𝟐=−𝟑 𝒙𝒙=
𝟐𝟑
𝒙=−𝟐𝟑
Factor by taking out common factor – 1
−𝟏 (𝟗 𝒙¿¿𝟐−𝟒)¿−𝟏 (𝟑 𝒙 −𝟐)(𝟑 𝒙+𝟐)
Solve = 0
−𝟗 𝒙𝟐=−𝟒
Common factor – 1 is negative.Graph opens down.
(−𝟐𝟑
, 0) (𝟐𝟑
, 0)
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 𝑥5 −8 𝑥4+6 𝑥3
2 𝑥5 −8 𝑥4+6 𝒙𝟑
2 𝒙𝟑(𝑥¿¿5 − 3− 4 𝑥4 −3+3 𝑥𝟑− 𝟑)¿
2 𝑥3(𝑥¿¿2 − 4 𝑥1+3 𝑥𝟎)¿
2 𝑥3(𝑥¿¿2 − 4 𝑥+3)¿
2 𝑥3(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 −1 8 𝑥− 2+6 𝑥− 3
−3 𝑥−1 −1 8 𝑥− 2+6 𝒙− 𝟑
−3 𝒙−𝟑(𝑥¿¿−1 − (− 3 )+6 𝑥−2 − (− 3) −2 𝑥− 𝟑−(−𝟑))¿
−3 𝒙−𝟑(𝑥¿¿𝟐+6𝑥𝟏− 2𝑥𝟎)¿
−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−8+6 𝑥−1
2 𝑥1 −8 𝒙𝟎+6 𝒙−𝟏
2 𝒙− 𝟏(𝑥¿¿1 −(−1)− 4 𝑥0 −(−1 )+3 𝑥− 𝟏−(−𝟏))¿
2 𝒙− 𝟏(𝑥¿¿ 2− 4 𝑥1+3 𝑥𝟎)¿
2 𝑥− 1(𝑥¿¿2 − 4 𝑥+3)¿
2 𝑥− 1(𝑥−3)(𝑥− 1)
13
12𝑥73 −24 𝑥
43 −36 𝑥
13
When subtracting rational exponents use a common denominator.
12𝑥73 −24 𝑥
43 −36 𝒙
𝟏𝟑
12 𝒙𝟏𝟑 (𝑥
𝟕𝟑
− 𝟏𝟑 −2 𝑥
𝟒𝟑
− 𝟏𝟑 −3 𝑥
𝟏𝟑
− 𝟏𝟑 )
12 𝒙𝟏𝟑 (𝑥
𝟔𝟑 −2 𝑥
𝟑𝟑 −3 𝑥𝟎)
12 𝒙𝟏𝟑 (𝑥𝟐−2 𝑥𝟏 −3 𝑥𝟎)
12 𝒙𝟏𝟑 (𝑥𝟐−2 𝑥−3)
12𝑥13 (𝑥− 3)(𝑥+1)
14
3 𝑥32 − 27 𝑥
− 12
When subtracting rational exponents use a common denominator.
3 𝑥32 − 27 𝒙
− 𝟏𝟐
3 𝒙− 𝟏
𝟐 (𝑥𝟑𝟐
− − 𝟏𝟐 −9 𝑥
− 𝟏𝟐
− −𝟏𝟐 )
3 𝒙− 𝟏
𝟐 (𝑥𝟒𝟐 − 9𝑥𝟎)
3 𝑥− 1
2 (𝑥2 − 9)
3 𝑥− 1
2 (𝑥−3)(𝑥+3)
15
4(x – 5)4 – 6(x – 5)3
4(x – 5)4 – 6(x – 5)3
2(x – 5)3 [2(x – 5)4-3 – 3(x – 5)3-3]
2(x – 5)3 [2(x – 5)1 – 3(x – 5)0]
2(x – 5)3 [2(x – 5) – 3]
2(x – 5)3 [2x – 10 – 3]
2(x – 5)3 (2x – 13)
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 )
− 32
−( −32 )
]
2 ( 𝒙+𝟔 )− 3
2 [ 4 ( 𝒙+𝟔 )22 − 3 ( 𝒙+𝟔 )0]
2 ( 𝒙+𝟔 )− 3
2 [ 4 ( 𝒙+𝟔 )1− 3 ( 𝒙+𝟔 )0]
2 ( 𝒙+𝟔 )− 3
2 [ 4 𝑥+24−3 ]
2 (𝑥+6 )− 3
2 [4 𝑥+21]
18
Factor by Decomposition Example 6x2 – 11x + 3
𝟔×𝟑=182 ×− 9=18
2+(− 9)=−7
6 𝑥2− 11𝑥+3
6 𝑥2+2𝑥−9 𝑥+3
2 𝑥(3𝑥+1)− 3(3𝑥+1)
(2 𝑥−3)(3𝑥+1)
19
Quadratic Formula ax2 + bx + c = 0
𝒙=−𝒃±√𝒃𝟐−𝟒𝒂𝒄𝟐𝒂
20
Solve for x 3x2 – 2x – 4 = 0
𝑥=−(−2)±√(−2)2− 4 (3)(− 4)
2(3)
𝑥=2 ±√4+486
𝑥=2 ±√526
𝑥=2 ±√4 ×136
𝑥=2 ±2√136
Answers in simplestand exact radical formApproximate decimal answers
to nearest hundredth.
21
𝑥=−(−3)±√(− 3)2− 4 (5)(10)
2(5)
𝑥=3 ±√9 −20010
𝑥=3 ±√−1916
Non-real answer.
Solve for x 5x2 – 3x + 10 = 0
22
SYNTHETIC DIVISION
Method I: SUBTRACTION
– 3 1 4 – 5 –12
– 3 –21 –48
1 7 16 36
Divide x3 + 4x2 – 5x – 12 by x – 3
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
Method II: ADDITION
3 1 4 – 5 –12
3 21 48
1 7 16 36
Divide x3 + 4x2 – 5x – 12 by x – 3
Quotient is x2 + 7x + 16
Remainder is 36
NOTE: x3 + 4x2 – 5x 12
(3)3 + 4(3)2 – 5(3) – 12 = 36
24
SYNTHETIC DIVISION
Divide x3 + 3x2 – 5 by x + 2
Method I: SUBTRACTION
+ 2 1 3 0 –5
2 2 –4
1 1 –2 –1
Quotient is x2 + x – 2
Remainder is –1
NOTE: x3 + 3x2 – 5
(–2)3 + 3(–2)2 – 5 = –1
𝒙𝟑+𝟑𝒙 𝟐+𝟎 𝒙 −𝟓
25
SYNTHETIC DIVISION
Divide x3 + 3x2 – 5 by x + 2
Method II: ADDITION
- 2 1 3 0 –5
–2 –2 4
1 1 –2 –1
Quotient is x2 + x – 2
Remainder is –1
NOTE: x3 + 3x2 – 5
(–2)3 + 3(–2)2 – 5 = –1
26
SYNTHETIC DIVISION
Divide x3 – 8 by x – 2
Method I: SUBTRACTION
– 2 1 0 0 –8
–2 –4 –8
1 2 4 0
Quotient is x2 + 2x + 4
Remainder is 0
NOTE: x3– 8
(2)3 – 8 = 0
𝒙𝟑+𝟎 𝒙𝟐+𝟎 𝒙 −𝟖
27
Factorx3 – 8
Difference of Cubes Formulaa3 – b3 = (a – b)(a2 + ab + b2)
Factor27x3 – 64
𝑎=𝑥 𝑏=2(𝑥− 2)(𝑥2+2 𝑥+22)
(𝑥− 2)(𝑥2+2 𝑥+4)
If we compare this answer to the previous slide we see it is the same.This is a shortcut that will help withmore difficult questions.
𝑎=3𝑥 𝑏=4
(3 𝑥− 4 )[ (3 𝑥 )2+(3 𝑥)(4)+42]
(3 𝑥− 4 )(9 𝑥2+12𝑥+16)
28
SYNTHETIC DIVISION
Divide x3 + 27 by x + 3
Method I: SUBTRACTION
+3 1 0 0 27
3 –9 27
1 –3 9 0
Quotient is x2 – 3x + 9
Remainder is 0
NOTE: x3+ 27
(–3)3 + 27 = 0
29
Factorx3 + 27
Sum of Cubes Formulaa3 + b3 = (a + b)(a2 – ab + b2)
Factor
𝑎=𝑥 𝑏=3(𝑥+3)(𝑥2−3 𝑥+32)
(𝑥+3)(𝑥2−3 𝑥+9)
If we compare this answer to the previous slide we see it is the same.This is a shortcut that will help withmore difficult questions.
𝑎=3𝑥𝑏=4
(3 𝑥− 4 )[ (3 𝑥 )2+(3 𝑥)(4)+(4)2]
(3 𝑥− 4 )(9 𝑥2+12𝑥+16)
Factor𝑎=𝑥 𝑏=5 𝑦
(𝑥+5 𝑦 )[ (𝑥 )2−(𝑥)(5 𝑦)+(5 𝑦 )2]
(𝑥+5 𝑦 )(𝑥2− 5𝑥𝑦+25 𝑦2)