IPMAT Indore 2023 · Quantitative Ability · Question 8

Step 1 — apply the Remainder Theorem. Dividing p\left( x \right) by \left( x-1 \right) leaves remainder p\left( 1 \right). So

p\left( 1 \right)=2

Step 2 — evaluate p\left( 1 \right). At x=1 every power of x equals 1, so p\left( 1 \right) is simply the sum of the coefficients:

4-1+3-5+c+2-1+1-4+6-2

Add the known numbers in order: 4-1=3; +3=6; -5=1; +2=3; -1=2; +1=3; -4=-1; +6=5; -2=3.

p\left( 1 \right)=3+c

Step 3 — solve.

3+c=2 \Rightarrow c=-1

Step 4 — answer what was asked.

c+6=-1+6=5

Answer: 5

Shortcut. “Divided by x-1” means “add up the coefficients”. That is the entire question — the ten-term polynomial is there to waste your time, not to be divided.

Sum the knowns to get 3, so 3+c=2 and c=-1.

Then read the question again. It asks for c+6, not c — answer 5. Questions that ask for a shifted or scaled version of the unknown are common in the SA section precisely because candidates stop one step early; circle what is being asked before you start.