Question

What must be added to 2x⁴ – 5x³ + 2x² – x – 3 so that the result is exactly divisible by (x – 2)?

Answer 1

Let k be added to 2x⁴ – 5x³ + 2x² – x – 3 so that the result is exactly divisible by (x – 2). Here, k is a constant.

∴ f(x) = 2x⁴ – 5x³ + 2x² – x – 3 + k is exactly divisible by (x – 2).

Using factor theorem, we have

$$\begin{gathered} f\left(2\right)=0 \\ \Rightarrow 2\times 2^4-5\times 2^3+2\times 2^2-2-3+k=0 \\ \Rightarrow 32-40+8-5+k=0 \\ \Rightarrow -5+k=0 \\ \Rightarrow k=5 \end{gathered}$$
Thus, 5 must be added to 2x⁴ – 5x³ + 2x² – x – 3 so that the result is exactly divisible by (x – 2).