Question

Plzz help me in solving q.no.8

Answer 1

Hello there! Please see the picture file I have attached to this answer.

First, let's take  $p(x) = x^3 + (3m+1) x^2 +nx-18$.

The question states that both $(x-1)$ and $(x+2)$  are factors of $p(x)$. So, when we apply Remainder Theorem, we get the following equations:

1. $p(+1)=0$
2. $p(-2)=0$

When you evaluate the above two equations, you get the following (in order):

1. $3m+n=16$
2. $6m-n=11$

Now, you may remember how to solve a pair of linear equations of two variables by Substitution or Elimination method. You have to do it here as well. I will try the Elimination method, as shown in the attached picture.

When you do that, you get that $9m=27$, hence $m=3$.

Use this value of $m$ and substitute in any one equation, as shown in the picture. Upon doing so, you get that $n=7$.

That's all! It's very easy, and you get $m=3$, and $n=7$!

Concepts to apply/remember:

1. Remainder Theorem
2. Solving a pair of Linear Equations of Two Variables