You can put this solution on YOUR website! Given:
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Rearrange the terms in the numerator to get:
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Then factor -1 from the two terms in the numerator to get:
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Now note that the terms inside the parentheses of the numerator are the difference of two
terms that are squares. The factor form for this difference of squares is:
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Applying this rule to the terms inside the parentheses of the numerator results in:
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Next let's work on the denominator. Notice that each term in the denominator contains
the variable x. So factor out an x and the denominator becomes:
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Now we need to factor the quadratic in the parentheses. Since the multiplier of the term is 1, we know that the factors of the quadratic will be of the form:
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(x _ ___)*(x _ ___)
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where the underlines in each set of parentheses represent a missing sign and a missing
number.
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The two missing numbers (one in each set of parentheses) must be factors of 8 so that when
they are multiplied together their product is 8. One of them must be minus and the other
must be plus because the product of these two factors must be -8. We also know that their
sum must be -2 which is needed for the middle term of the quadratic in the denominator.
So we are looking for factors that multiply to make -8 and add to make -2.
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The factor pairs of 8 are (8 and 1) and (4 and 2). There's no way that 8 and 1 can add
to be -2, but if we made the factor pair -4 and +2, then the product would be -8 and the
sum would be -2. So we know the factors of the quadratic in the denominator are:
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Substituting this into the factored denominator now makes the entire problem:
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and canceling the common terms in the numerator and denominator:
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results in:
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That's the answer to the problem.
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Hope this helps you to understand the process of simplifying by factoring and canceling
like terms in the numerator and the denominator.
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