document.write( "Question 714127: A curve has as its equation y = x3 - kx2 - 16x + 32\r
\n" ); document.write( "\n" ); document.write( "You then find that k = 2\r
\n" ); document.write( "\n" ); document.write( "*Tricky part* The point B(p,35) also lies on this curve. Find the Value of p.
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\n" ); document.write( "From the given graph it looks like p = about -4 but I cannot find a way to work it out.\r
\n" ); document.write( "\n" ); document.write( "Thank you for your help, Andrew \n" ); document.write( "

Algebra.Com's Answer #438691 by jsmallt9(3758) $\"\"$ $\"About$

You can put this solution on YOUR website!
With k = 2 the equation becomes:
\n" ); document.write( " $\"y+=+x%5E3+-+2x%5E2+-+16x+%2B+32\"$
\n" ); document.write( "If point B lies on the curve then its coordinates must fit the equation. So:
\n" ); document.write( " $\"35+=+p%5E3+-+2p%5E2+-+16p+%2B+32\"$
\n" ); document.write( "Now we just solve this for p.

\n" ); document.write( "The first thing we must do is get one side to be zero. Subtracting 35 we get:
\n" ); document.write( " $\"0+=+p%5E3+-+2p%5E2+-+16p+-+3\"$
\n" ); document.write( "Next we try to factor the right side. The greatest common factor (GCF) is 1 (which we rarely bother factoring out). There are too many terms for factoring by patterns or for trinomial factoring. And I do not see how to factor by grouping. All that's left is trial and error of the possible rational roots.

\n" ); document.write( "The possible rational roots of a polynomial are all the possible ratios, positive and negative, made with a factor of the constant term (at the end) over a factor of the leading coefficient (at the front). Our constant term is 3 whose factors are 1 and 3. (Actually it is -3 but since we will try all positive and negative ratios we can just as well use 3.) And the leading coefficient (in front $\"p%5E3\"$ ) is 1 whose factors are just 1's. This makes the possible rational roots:
\n" ); document.write( "+1/1 and +3/1
\n" ); document.write( "which simplify to:
\n" ); document.write( "+1 and +3
\n" ); document.write( "So there are 4 possible rational roots.

\n" ); document.write( "Since you've already figured out that it should be near -4, we'll try -3 first. Checking to see if a possible root is a root is more easily done with synthetic division:
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document.write( "-3 |   1   -2   -16   -3\r\n" );
document.write( "----       -3    15    3\r\n" );
document.write( "      ------------------\r\n" );
document.write( "       1   -5    -1    0\r\n" );
document.write( "

The remainder, in the lower right corner, is zero. This means that (x - (-3)) is a factor and that -3 is a root of the polynomial. So p = -3 or the coordinates of B are (-3. 35).
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