Question 128546
Given to simplify:
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{{{(1-3x)/(3x^2+14x-5)}}}
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Since these are "setup" problems that are designed to teach a point, you can make a guess
that the numerator and the denominator have a common term that cancels out.
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Let's re-write the numerator that it is arranged in descending powers of x ... that is
so the x term appears first followed by the constant. When we do that the problem becomes:
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{{{(-3x+1)/(3x^2+14x-5)}}}
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Then in the numerator, factor out {{{-1}}} so that we make the {{{3x}}} term positive. This makes the
expression become:
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{{{(-1)(3x-1)/(3x^2+14x-5)}}}
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Now let's see if we can factor the denominator. And let's guess that one of the factors
might be {{{3x-1)}}}. This being the case we might begin the factoring with:
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{{{(3x -1)*(Ax + B)}}}
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We know that {{{3x}}} times {{{Ax}}} has to equal the {{{3x^2}}} of the given denominator. Therefore,
{{{A}}} will have to equal {{{1}}} so that {{{3x*1x = 3x^2}}}. So go to our factored form of the denominator
and substitute {{{1}}} for {{{A}}} to make the factored form:
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{{{(3x-1)*(1x + B)}}}
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Note that when we do the next step we just write the {{{1x}}} as {{{x}}}
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Next from this factored form we can see that if we multiply the constant {{{-1}}} by
B, it has to result in the {{{-5}}} of the original denominator of the problem. So we can 
say that:
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{{{(-1)*B = -5}}}
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and we can solve this for B by multiplying both sides by {{{-1}}} to get:
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{{{B = 5}}}
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Substitute this value for B in the factored form of our denominator and you have that the
factored form is:
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{{{(3x-1)*(x + 5)}}}
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You can check this out by doing the multiplication just to be sure that the product is
equal to the original denominator.
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Substitute this into the problem where we had rearranged the numerator and we now have:
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{{{(-1)(3x-1)/((3x-1)*(x+5))}}}
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Cancel the common factor in the numerator with the same factor in the denominator:
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{{{(-1)*(cross(3x-1))/((cross(3x-1))*(x+5))}}}
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And you are left with the simplified form:
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{{{-1/(x+5)}}}
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and that's your answer. 
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Hope this explanation helps you to understand the problem.
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