U1A L1 Examples FACTORING REVIEW EXAMPLES

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)

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

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)

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

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

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)

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

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.

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)

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.

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) )

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

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

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]

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)

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 2 2 −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]

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)

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

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± 4+48 6 𝑥= 2± 4×13 6 𝑥= 2±2 13 6 𝑥= 2± 52 6 𝑥= 1± 13 3 Answers in simplest and exact radical form Approximate decimal answers to nearest hundredth. 𝑥=−0.95 𝑎𝑛𝑑 𝑥=1.54

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

SYNTHETIC DIVISION Divide x3 + 4x2 – 5x – 12 by x – 3 Method I: SUBTRACTION – 3 1 4 – 5 –12 – 3 –21 –48 1 7 16 36 Quotient is x2 + 7x + 16 Remainder is 36 NOTE: x3 + 4x2 – 5x 12 (3)3 + 4(3)2 – 5(3) – 12 = 36

SYNTHETIC DIVISION Divide x3 + 4x2 – 5x – 12 by x – 3 Method II: ADDITION 3 1 4 – 5 –12 3 21 48 1 7 16 36 Quotient is x2 + 7x + 16 Remainder is 36 NOTE: x3 + 4x2 – 5x 12 (3)3 + 4(3)2 – 5(3) – 12 = 36

Divide x3 + 3x2 – 5 by x + 2 SYNTHETIC DIVISION 𝒙𝟑 + 𝟑𝒙𝟐 + 𝟎𝒙 −𝟓 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

Divide x3 + 3x2 – 5 by x + 2 SYNTHETIC DIVISION 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

Divide x3 – 8 by x – 2 SYNTHETIC DIVISION 𝒙𝟑 +𝟎 𝒙 𝟐 +𝟎𝒙−𝟖 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

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)

Divide x3 + 27 by x + 3 SYNTHETIC DIVISION 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

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 )