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FACTORING REVIEW EXAMPLES 1

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Factor x 2 + 3x – 4Solve x 2 + 3x – 4 = 0 Graph Y 1 = x 2 + 3x – 4 Find x-intercepts What _____× _____ = – 4 and _____+ _____ = 3 2

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Factor x 2 + 3x + 2 What _____× _____ = 2 and _____+ _____ = 3 Factor 2x 2 + 6x + 4 by taking out common factor 2 Factor –3x 2 – 9x – 6 by taking out common factor – 3 3 2(x 2 + 3x + 2) – 3(x 2 + 3x + 2) 2(x 2 + 3x + 2) = 0 – 3(x 2 + 3x + 2) = 0

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Graph Y 1 = x 2 + 3x + 2 Y 2 = 2x 2 + 6x + 4or Y 2 = 2(x 2 + 3x + 2) Y 3 = –3x 2 – 9x – 6 orY 3 = –3(x 2 + 3x + 2) 4 Find x-intercepts

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Factor –x 2 – 6x – 8 by taking out common factor –1 Solve –x 2 – 6x – 8 = 0 Graph Y 1 = –x 2 – 6x – 8 5 – (x 2 + 6x + 8) – (x 2 + 6x + 8) = 0 Find x-intercepts

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Can you factor x ? Can you solve x = 0 Graph Y 1 = x 2 – 4 6 NO Non-Real Answer Find x-intercepts There are NO x - intercepts

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Graph Y 1 = x 2 – 4 7 Find x-intercepts Difference of Squares Using this method it VERY easy to forget BOTH answers!!!! OR

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8 Factor 8x 2 – 18 by taking out common factor 2 Solve 8x 2 – 18 = 0 Common factor 2 is positive. Graph opens up.

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9 Common factor – 1 is negative. Graph opens down.

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10 MORE COMMON FACTORING EXAMPLES When dividing out common factors look for the greatest common numerical factor and the smallest exponent on the variables.

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11 MORE COMMON FACTORING EXAMPLES When dividing out common factors look for the greatest common numerical factor and the smallest exponent on the variables. This example will NOT factor further.

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12 MORE COMMON FACTORING EXAMPLES When dividing out common factors look for the greatest common numerical factor and the smallest exponent on the variables.

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13 When subtracting rational exponents use a common denominator.

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14 When subtracting rational exponents use a common denominator.

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15 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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18 Factor by Decomposition Example 6x 2 – 11x + 3

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19 Quadratic Formula ax 2 + bx + c = 0

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20 Solve for x 3x 2 – 2x – 4 = 0

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21 Non-real answer. Solve for x 5x 2 – 3x + 10 = 0

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

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

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

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

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

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27 Factor x 3 – 8 Difference of Cubes Formula a 3 – b 3 = (a – b)(a 2 + ab + b 2 ) Factor 27x 3 – 64 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.

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28 SYNTHETIC DIVISION Divide x by x + 3 Method I: SUBTRACTION – –3 9 0 Quotient is x 2 – 3x + 9 Remainder is 0 NOTE: x (–3) = 0

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29 Factor x Sum of Cubes Formula a 3 + b 3 = (a + b)(a 2 – ab + 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.

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