If (x + 1) and (x - 2) are factors of the polynomial \(g(x) = x^{4} + ax^{3} + bx^{2} - 16x - 12\), find the values of a and b.

Assessment: WAEC SSCE - Further Mathematics - 2017 (Essay) Subject: Further Mathematics

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By the Factor Theorem, \(g(-1)=0\) and \(g(2)=0\).

\(g(x)=x^{4}+ax^{3}+bx^{2}-16x-12\).

Using \(g(-1)=0\):

\[1-a+b+16-12=0\ \Rightarrow\ -a+b+5=0\ \Rightarrow\ b=a-5\quad(1)\]

Using \(g(2)=0\):

\[16+8a+4b-32-12=0\ \Rightarrow\ 8a+4b-28=0\ \Rightarrow\ 2a+b=7\quad(2)\]

Substitute (1) into (2): \(2a+(a-5)=7\Rightarrow3a=12\Rightarrow a=4\).

Then \(b=4-5=-1\).

\[a=4,\qquad b=-1\]

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