Question 8, Review Exercise

Solutions of Question 8 of Review Exercise of Unit 05: Polynomials. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan.

If two linear factors of the polynomial $y^{3}+6 y^{2}-y-30$ are $(y-2)$ and $(y+3)$, find its third factor.

Solution.

Suppose $p(y)=y^{3}+6 y^{2}-y-30$.

Since $(y-2)$ and $(y+3)$ are factor of $p(y)$, this gives $2$ and $-3$ are zeros of $p(y)$.

Using synthetic division: \[ \begin{array}{r|rrrr} 2 & 1 & 6 & -1 & -30 \\ & \downarrow & 2 & 16 & 30 \\ \hline -3 & 1 & 8 & 15 & 0 \\ & \downarrow & -3 & -15 & \\ \hline & 1 & 5 & 0 & \\ \end{array} \]

Thus $$p(y)=(y-2)(y+3)(y+5).$$

Hence $y+5$ is the third factor.

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