document.write( "Question 695486: Show that y+1 is a factor of the expression y^3+(m-2)y^2+(m-7)y-4 for all values of m. If the expression also has a factor y+2, find the value of m and the third factor. \n" ); document.write( "

Algebra.Com's Answer #428514 by mouk(232) $\"\"$ $\"About$

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Let $\"+f%28y%29+=+y%5E3%2B%28m-2%29y%5E2%2B%28m-7%29y-4+\"$ \r
\n" ); document.write( "\n" ); document.write( "By the factor theorem $\"+y-A+\"$ is a factor of $\"+f%28y%29+\"$ if and only if $\"+f%28A%29=0+\"$ \r
\n" ); document.write( "\n" ); document.write( "So $\"+y%2B1+\"$ is a if and only if $\"+f%28-1%29+=+0+\"$ , Now
\n" ); document.write( " $\"+f%28-1%29+=+%28-1%29%5E3%2B%28m-2%29%28-1%29%5E2%2B%28m-7%29%28-1%29-4+\"$
\n" ); document.write( " $\"+f%28-1%29+=++-1+%2B+%28m-2%29+-+%28m-7%29+-+4+\"$
\n" ); document.write( " $\"+f%28-1%29+=++0+\"$
\n" ); document.write( "so $\"+f%28-1%29+=+0+\"$ showing $\"+y%2B1+\"$ is a factor QED\r
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\n" ); document.write( "\n" ); document.write( "Similarly, $\"+f%28-2%29=0+\"$ so,
\n" ); document.write( " $\"+%28-2%29%5E3+%2B+%28m-2%29%28-2%29%5E2+%2B+%28m-7%29%28-2%29+-+4+=+0+\"$
\n" ); document.write( " $\"+-8+%2B+4%28m-2%29+-2%28m-7%29+-+4+=+0+\"$
\n" ); document.write( " $\"+-8+%2B+4m-8+-2m+%2B14+-+4+=+0+\"$
\n" ); document.write( " $\"+2m+-+6+=0++\"$ giving $\"+m+=+3+\"$ \r
\n" ); document.write( "\n" ); document.write( "So substituting $\"+m+=+3+\"$ , gives:
\n" ); document.write( " $\"+f%28y%29+=+y%5E3+%2B+%283-2%29y%5E2+%2B+%283-7%29y+-+4+\"$
\n" ); document.write( " $\"+f%28y%29+=+y%5E3+%2B+y%5E2+-+4y+-+4+\"$ \r
\n" ); document.write( "\n" ); document.write( "We now know that both $\"+y+%2B+1+\"$ and $\"+y+%2B+2+\"$ are factors
\n" ); document.write( "Suppose the last factor is $\"+y%2Bn+\"$ then
\n" ); document.write( " $\"+f%28y%29+=+y%5E3+%2B+y%5E2+-+4y+-+4+\"$
\n" ); document.write( " $\"+f%28y%29+=+%28y%2B1%29%28y%2B2%29%28y%2Bn%29+\"$ \r
\n" ); document.write( "\n" ); document.write( "Multiplying out gives:
\n" ); document.write( " $\"+f%28y%29+=+%28y%5E2+%2B+3y+%2B+2%29%28y%2Bn%29+\"$
\n" ); document.write( " $\"+f%28y%29+=+y%5E3+%2B+3y%5E2+%2B+2y+%2B+ny%5E2+%2B+3ny+%2B+2n+\"$
\n" ); document.write( " $\"+f%28y%29+=+y%5E3+%2B+%283%2Bn%29y%5E2+%2B+%283n%2B2%29y+%2B+2n+\"$ \r
\n" ); document.write( "\n" ); document.write( "So Comparing with the constant coefficient of $\"+f%28y%29+=+y%5E3+%2B+y%5E2+-+4y+-+4+\"$
\n" ); document.write( "gives: $\"+2n+=+-4+\"$ so $\"+n=-2+\"$ \r
\n" ); document.write( "\n" ); document.write( "Hence the third factor is $\"+y-2+\"$ \r
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