FACTORING REVIEW EXAMPLES 1. Factor x 2 + 3x – 4Solve x 2 + 3x – 4 = 0 Graph Y 1 = x 2 + 3x –...

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FACTORING REVIEW

EXAMPLES

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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)

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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

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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

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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

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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

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(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

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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)

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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)

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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)

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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)

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−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.

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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)

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12𝑥73 −24 𝑥

43 −36 𝑥

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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)

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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)

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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)

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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)

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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]

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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)

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Quadratic Formula ax2 + bx + c = 0

𝒙=−𝒃±√𝒃𝟐−𝟒𝒂𝒄𝟐𝒂

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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.

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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

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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

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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


Remainder is 36

NOTE: x3 + 4x2 – 5x 12

(3)3 + 4(3)2 – 5(3) – 12 = 36

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SYNTHETIC DIVISION

Divide x3 + 3x2 – 5 by x + 2


+ 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

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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

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SYNTHETIC DIVISION

Divide x3 – 8 by x – 2


– 2 1 0 0 –8

–2 –4 –8

1 2 4 0


Remainder is 0

NOTE: x3– 8

(2)3 – 8 = 0

𝒙𝟑+𝟎 𝒙𝟐+𝟎 𝒙 −𝟖

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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)

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SYNTHETIC DIVISION

Divide x3 + 27 by x + 3


+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

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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)

FACTORING REVIEW EXAMPLES 1. Factor x 2 + 3x – 4Solve x 2 + 3x – 4 = 0 Graph Y 1 = x 2 + 3x –...

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