document.write( "Question 75935: can someone help me write to simplest form
\n" ); document.write( "4r^2-25s^2
\n" ); document.write( "over
\n" ); document.write( "2r^2+3rs-20s^2 \n" ); document.write( "
Algebra.Com's Answer #54531 by bucky(2189)\"\" \"About 
You can put this solution on YOUR website!
.
\n" ); document.write( "\"%284r%5E2+-+25s%5E2%29%2F%282r%5E2+%2B+3rs+-+20s%5E2%29\"
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\n" ); document.write( "The numerator of this fraction can easily be factored if you recognize that is is the difference
\n" ); document.write( "of two squared terms. The form is:
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\n" ); document.write( "\"%28x%5E2+-+y%5E2%29+=+%28x+%2B+y%29%2A%28x+-+y%29\"
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\n" ); document.write( "This says that if you have the difference of two squares you can factor it into the product
\n" ); document.write( "of the sum and difference of their square roots.
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\n" ); document.write( "Applying this rule to the numerator of the fraction we can see that the square root of
\n" ); document.write( "each of the terms in the numerator are 2r and 5s. So the factored form of the numerator
\n" ); document.write( "is \"%282r%2B5s%29%2A%282r-5s%29\" and the problem now takes the form:
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\n" ); document.write( "\"%28%282r%2B5s%29%2A%282r-5s%29%29%2F%282r%5E2+%2B+3rs+-+20s%5E2%29\"
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\n" ); document.write( "The following is a schoolwork suggestion. This problem was set up to teach you something.
\n" ); document.write( "It is likely that the lesson is to learn to cancel a factor in the denominator with a like
\n" ); document.write( "factor appearing in the numerator. Therefore, it likely that the denominator also contains
\n" ); document.write( "one of the factors that we found in the numerator. That's a clue to what we should look
\n" ); document.write( "for when we factor the denominator. So let's factor the denominator next ...
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\n" ); document.write( "We can tell from the first term in the denominator \"2r%5E2\" that it factors into
\n" ); document.write( "2r and r. So we know that our factored denominator is of the form:
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\n" ); document.write( "(2r ______ ) * (r ______ )
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\n" ); document.write( "The last term in the denominator \"-20s%5E2\" tells us something. Because it has a minus
\n" ); document.write( "sign, one of its factors must be positive and one must be negative. (If they both were
\n" ); document.write( "positive or both negative they would multiply together to give a positive, not a negative
\n" ); document.write( "term.) We also know that both the factors contain an s, so we can write the factored
\n" ); document.write( "form of the denominator as:
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\n" ); document.write( "(2r ____ s)*(r ____ s)
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\n" ); document.write( "All we need to do is find the positions of the + and - signs and the numbers for the blanks.
\n" ); document.write( "We can now guess that 5 is the number for the blank in the factor containing 2r because
\n" ); document.write( "that would make it possible to cancel that term with a like term in the numerator.
\n" ); document.write( "So we can guess that the factor is now of the form:
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\n" ); document.write( "(2r ___ 5s)*(r ___ s)
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\n" ); document.write( "Now we can tell that the number in the second factor must be 4. Why? Because the 5s at
\n" ); document.write( "the end of the first factor must multiply it and result in \"-20s%5E2\". So we can now
\n" ); document.write( "write our factored form of the denominator as:
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\n" ); document.write( "(2r ___ 5s)*(r ___ 4s)
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\n" ); document.write( "All we have to do now is to find the position of the + and - signs that go into the blanks.
\n" ); document.write( "We know that 2r times 4s and then r times 5s produce two products that add together
\n" ); document.write( "to give + 3rs of the original denominator. These products are 8rs and 5rs. For these terms
\n" ); document.write( "to combine to give +3s, we can see that 8rs must be positive and 5rs must be negative.
\n" ); document.write( ".
\n" ); document.write( "For the 8rs to be positive, the 4s must be positive. And therefore the 5s must be negative.
\n" ); document.write( "This means that the factors in the denominator are:
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\n" ); document.write( "(2r - 5s)*(r + 4s)
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\n" ); document.write( "[You can always check this by multiplying the two terms together to see if the result
\n" ); document.write( "really is the original form of the denominator in the problem.]
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\n" ); document.write( "Now substitute this pair of factors for the original denominator and the problem is then:
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\n" ); document.write( "\"%28%282r%2B5s%29%2A%282r-5s%29%29%2F%28%282r-5s%29%2A%28r%2B4s%29%29\"
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\n" ); document.write( "Next cancel the like terms in the denominator and numerator:
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\n" ); document.write( "\"%28%282r%2B5s%29%2A%28cross%282r-5s%29%29%29%2F%28%28cross%282r-5s%29%29%2A%28r%2B4s%29%29\"
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\n" ); document.write( "And you are left with the answer:
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\n" ); document.write( "\"%282r%2B5s%29%2F%28r%2B4s%29\"
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\n" ); document.write( "Hope this helps you to see how to simplify the original problem into its \"reduced\"
\n" ); document.write( "form. \n" ); document.write( "
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