Factoring Review
Geometry Practice with Factoring
Name _____________________________ posted October 25, 2003
Explanation:
When factoring, the sign of the last term determines if the signs are the same or different. If the last term is positive, both signs will be the same - either both are positive or both are negative. If the last term is negative, one of the factors will be a sum and the other will be a difference.
x² + 9x + 20
The sign with the 20 is positive, therefore, the factors will both be differences or both will be sums. In this case, since the 9 is positive, both factors will be sums.
Factors of +20
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Sum
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+20 +1
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21
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+10 +2
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12
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+ 5 +4
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9 (This is what we want to use.)
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x² + 9x + 20
x² + 5x + 4x + 20
(x² + 5x) + (4x + 20)
x(x + 5) + 4 (x + 5)
(x + 5) (x + 4)
x² - 9x + 20
The sign with the 20 is positive, therefore, the factors will both be differences or both will be sums. In this case, since the 9 is negative, both factors will be negative.
Negative Factors of 20
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Sum
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-20 -1
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-21
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-10 -2
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-12
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- 5 -4
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-9 (This is what we want to use.)
|
x² - 9x + 20
x² - 5x - 4x + 20
(x² - 5x) + (-4x + 20)
x(x - 5) - 4(x - 5)
(x - 5)(x - 4)
x² + 8x - 20 The last term is negative, so one factor will be positive and the other will
be negative. The middle term is positive, so the largest term will be
positive.
Factors of -20
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Sum
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20 -1
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19
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10 -2
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8 (This is what we want)
|
x² + 8x - 20
x² + 10x - 2x - 20
(x² + 10x) + (- 2x -20)
x (x + 10) -2(x + 10)
(x + 10)(x - 2)
2x² + 13x + 20
Both factors will be positive because the last term, 20, is positive, and the middle term, 13, is positive.
Multiply the coefficient of the first term by the constant (the last term). (2)(20) = 40
Factors of 40
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Sum
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40 1
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41
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20 2
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22
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10 4
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14
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8 5
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13 (This is what we want)
|
2x² + 13x + 20
2x² + 8x + 5x + 20
2x (x + 4) + 5(x + 4)
(x + 4) (2x + 5)
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2x² + 13x + 20
2x² + 5x + 8x + 20
x (2x + 5) + 4(2x + 5)
(2x + 5) (x + 4)
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Notice, the order of middle terms do not change the factors.
Use this method to factor the following:
1. x² - 2x - 8
2. 3x² + 10x + 3 (Multiply the first coefficient and the last coefficient to determine the needed product. Ask yourself what two numbers have a sum of 10 and a product of 3(3) or 9.)
3. v² - 14v + 49 4. x² - 6x + 25
5. x² + 6x + 8 6. x² + 10x + 25
7. 2x² - 2x - 24 8. 3x² - 8x + 4
Difference of Squares
Explanation: x² - 25 is a difference or a sum of negative numbers. The x is a perfect square and the 25 is a perfect square. This is called the difference of squares. The factors are easy. Take the square root of each term (x² - 25 is really x² + (-25) so these are terms). One factor is a sum and the other is a difference.
Examples
x² - 25 4x² - 36 81x² - 16
(x + 5) (x - 5) (2x - 6) (2x + 6) (9x - 4) (9x + 4)
Factor:
9. x² - 1 10. 25x² - 121
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