The polynomial f(x) =2x\(^3\) + px+ qx - 5 has (x-1) as a factor and a remainder of 27 when divided by (x + 2), where p and q are constants. Find the values...
Question 1 Report
The polynomial f(x) =2x\(^3\) + px+ qx - 5 has (x-1) as a factor and a remainder of 27 when divided by (x + 2), where p and q are constants. Find the values of p and q.
Answer Details
Reading the polynomial as \(f(x)=2x^{3}+px^{2}+qx-5\) (the two linear terms in the printed stem are a typographical slip for a quadratic and a linear term).
Since \((x-1)\) is a factor, \(f(1)=0\):
\[2+p+q-5=0\Rightarrow p+q=3\quad(\text{i})\]
The remainder is \(27\) when divided by \((x+2)\), so \(f(-2)=27\):
\[2(-8)+p(4)+q(-2)-5=27\Rightarrow -16+4p-2q-5=27\]
\[4p-2q=48\Rightarrow 2p-q=24\quad(\text{ii})\]
Add (i) and (ii): \(3p=27\Rightarrow p=9\). Then \(q=3-9=-6\).
\[\boxed{p=9,\ q=-6}\]
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