document.write( "Question 1056898: Find a polynomial of degree 4 that has zeros -3, 2, -2 and -1 and with the coefficient of x^2 is -3. \n" ); document.write( "

Algebra.Com's Answer #671962 by math_helper(2461) $\"\"$ $\"About$

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General form of a 4th degree polynomial is:\r
\n" ); document.write( "\n" ); document.write( " $\"ax%5E4+%2B+bx%5E3+%2B+cx%5E2+%2B+dx+%2B+e++=+0+\"$ \r
\n" ); document.write( "\n" ); document.write( "This can be constructed from the zeros:
\n" ); document.write( "
\n" ); document.write( "The zero at -3 means there is a factor (x+3) (because plugging in x=-3 gives zero)
\n" ); document.write( "The zeros at 2,-2, and -1 yield three more factors: (x-2)(x+2) and (x+1), respectively.\r
\n" ); document.write( "\n" ); document.write( "Putting all the factors together:
\n" ); document.write( " (x+3)(x-2)(x+2)(x+1)
\n" ); document.write( " = (x+3)(x+1)(x+2)(x-2) (reordered, will multiply first two, last two next)
\n" ); document.write( " = $\"%28x%5E2%2B4x%2B3%29%2A%28x%5E2-4%29\"$
\n" ); document.write( " = $\"x%5E4%2B4x%5E3%2B3x%5E2-4x%5E2-16x-12\"$
\n" ); document.write( " = $\"x%5E4%2B4x%5E3-x%5E2-16x-12\"$ \r
\n" ); document.write( "\n" ); document.write( " The problem said the coefficient of $\"x%5E2\"$ should be -3, since it is already -1, we need to multiply the polynomial by 3:\r
\n" ); document.write( "\n" ); document.write( " Ans: $\"3x%5E4%2B12x%5E3-3x%5E2-48x-36\"$ \n" ); document.write( "

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