Question 4 and 5, Exercise 5.1

Solutions of Question 4 and 5 of Exercise 5.1 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 $4 y^{3}-4 y^{2}+10+2 y$ is completely divisible by any of its factor such that the quotient is $4 y^{2}-8 y+10$, then find other factor.

Solution.

Find the value of ' $q$ ' if $x^{3}+q x^{2}-7 x+6$ is exactly divisible by $(x+1)$.

Solution.

Let $p(x)=x^{3}+q x^{2}-7 x+6$ and $x-c=x+1$ $\implies c=-1$.

By factor theorem $x+1$ is factor of $p(x)$ iff $p(-1)=0$.

This gives \begin{align*} &(-1)^3+q(-1)^2-7(-1)+6=0 \\ -&1+q+7+6=0\\ &q+12=0\\ &q=-12 \end{align*}

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