NIMCET 2021 — Mathematics PYQ

Finding the Value of p 
A polynomial $P(x)$ is exactly divisible by $(x-a)$ if and only if $P(a)=0$, according to the Factor Theorem. 
Step-by-step Solution 
1. Identify the divisor and its root: The polynomial $x^4-3x^3+2p{x}^2-6$ is exactly divisible by $(x-1)$. Therefore, the root of the divisor is $x=1$. 
2. Substitute the root into the polynomial: The value $x=1$ is substituted into the given polynomial
$(1{)}^4-3(1{)}^3+2p(1{)}^2-6=0$  
1. Simplify the equation: The equation is simplified by performing the calculations:
$1-3+2p-6=0$ $-2+2p-6=0$ $2p-8=0$ 
1. Solve for p: The equation is solved for $p$: 
$2p=8$ $p=\frac{8}{2}$ $p=4$ 
Final Answer 
The value of $p$ for which the polynomial $x^4-3x^3+2p{x}^2-6$ is exactly divisible by $(x-1)$ is $4$.