document.write( "Question 409239: Simplify using difference of squares:
\n" ); document.write( "(u^2+v-w)(u^2-v+w) \n" ); document.write( "

Algebra.Com's Answer #288154 by jsmallt9(3758) $\"\"$ $\"About$

You can put this solution on YOUR website!
$\"%28u%5E2%2Bv-w%29%28u%5E2-v%2Bw%29\"$
\n" ); document.write( "The difference of squares pattern is:
\n" ); document.write( " $\"%28a%2Bb%29%28a-b%29+=+a%5E2+-+b%5E2\"$
\n" ); document.write( "To use this pattern on your expression, we need to be able to rewrite your expression (at least in our heads) as the sum of an \"a\" and a \"b\" times the difference of that \"a\" and \"b\". Once you realize that the \"a\" and \"b\" can be any Math expression, then you will become a more powerful user of patterns.

\n" ); document.write( "Here's how we can rewrite your expression:
\n" ); document.write( " $\"%28u%5E2%2B%28v-w%29%29%28u%5E2-%28v-w%29%29\"$
\n" ); document.write( "Take a moment to see how this expression is equal to your original expression. Note how the minus in front of the parentheses in the second factor make the \"-w\" inside the parentheses equal to the \"+w\" in your original expression.

\n" ); document.write( "Once written this way, it is not hard to see that we have matched the difference of squares pattern (the left side) with the \"a\" being $\"u%5E2\"$ and the \"b\" being (v-w). So we can use the pattern to multiply, knowing that the answer will be difference of the squares of the \"a\" and the \"b\":
\n" ); document.write( " $\"%28u%5E2%29%5E2+-+%28v-w%29%5E2\"$
\n" ); document.write( "Squaring $\"u%5E2\"$ is simple. To square (v-w) we can use another pattern: $\"%28a-b%29%5E2+=+a%5E2+-2vw+%2B+w%5E2\"$ :
\n" ); document.write( " $\"u%5E4+-+%28v%5E2+-2vw+%2B+w%5E2%29\"$
\n" ); document.write( "Note the use of parentheses. That whole entire expression is $\"%28v-w%29%5E2\"$ . And if we were subtracting $\"%28v-w%29%5E2\"$ before then we need to subtract the whole expression when we replace $\"%28v-w%29%5E2\"$ .
\n" ); document.write( "One last simplification:
\n" ); document.write( " $\"u%5E4+-+v%5E2+%2B+2vw+-+w%5E2%29\"$

\n" ); document.write( "The alternative to using the pattern to multiply is to multiply the trinomials the \"normal\" way: Multiply each term of one polynomial by each term of the other and then add like terms,if any. This would mean 9 multiplications plus adding like terms. Using the patterns, once you learn how, makes this much easier. \n" ); document.write( "

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