document.write( "Question 271266: What is 36-13x^2+x^4? \n" ); document.write( "

Algebra.Com's Answer #198628 by jsmallt9(3758) $\"\"$ $\"About$

You can put this solution on YOUR website!
The direct answer to your question is that it is a polynomial expression.

\n" ); document.write( "But I'm guessing that you are supposed to factor this expression.. First let's get it in proper order, from highest exponent term to lowest:
\n" ); document.write( " $\"x%5E4-13x%5E2%2B36\"$
\n" ); document.write( "The easiest way to factor this is requires that you recognize that $\"x%5E4+=+%28x%5E2%29%5E2\"$ . That makes expression a quadratic in $\"x%5E2\"$ . It may help to see this and factor it if we use a temporary variable:
\n" ); document.write( "Let $\"q+=+x%5E2\"$ . Then $\"q%5E2+=+%28x%5E2%29%5E2+=+x%5E4\"$ and our expression becomes:
\n" ); document.write( " $\"q%5E2+-13q+%2B+36\"$
\n" ); document.write( "Factoring this trinomial is a matter of figuring out what factors of 36 add up to -13. There are many factors of 36. But since 36 is positive the factors we use must be both positive or both negative. And since we want them to add up to a negative number, we need two negative factors of 36. With some thought and trial and error we should find that -4 and -9 are the only factors of 36 that add up to -13. So our trinomial factors into:
\n" ); document.write( " $\"%28q+-+4%29%28q-13%29\"$
\n" ); document.write( "We are finished with our temporary variable so we will replace it with $\"x%5E2\"$ :
\n" ); document.write( " $\"%28x%5E2+-+4%29%28x%5E2+-+13%29\"$
\n" ); document.write( "When factoring you keep going until you cannot factor any further. The first factor is a difference of squares so it will factor according to the pattern: $\"a%5E2-b%5E2+=+%28a%2Bb%29%28a-b%29\"$ . Using this on the first factor we get:
\n" ); document.write( " $\"%28x%2B2%29%28x-2%29%28x%5E2-13%29\"$
\n" ); document.write( "which, if the problem was to factor the expression, is probably the desired answer.

\n" ); document.write( "If one really wants to go further with the factoring one could consider the last factor also a difference of squares (where $\"13+=+%28sqrt%2813%29%29%5E2\"$ ) and factor it according to the pattern, too:
\n" ); document.write( " $\"%28x%2B2%29%28x-2%29%28x+%2B+sqrt%2813%29%29%28x+-+sqrt%2813%29%29\"$

\n" ); document.write( "Eventually you may learn how to do without the temporary variable. IOW, go straight from
\n" ); document.write( " $\"x%5E4-13x%5E2%2B36\"$
\n" ); document.write( "to
\n" ); document.write( " $\"%28x%5E2+-+4%29%28x%5E2+-+13%29\"$ \n" ); document.write( "

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