Question

If the polynomial f(x) = 5x⁵ – 3x³ + 2x² – k gives remainder 1 when divided by x + 1, then k = __________.

Answer 1

Let f(x) = 5x⁵ – 3x³ + 2x² – k

To find the remainder obtained when 5x⁵ – 3x³ + 2x² – k is divided by x + 1,

we use remainder theorem, put x + 1 = 0.

f(−1) is the remainder.

Now,

$$\begin{gathered} f\left(-1\right)=5{\left(-1\right)}^5-3{\left(-1\right)}^3+2{\left(-1\right)}^2-k \\ \Rightarrow 1=5\left(-1\right)-3\left(-1\right)+2\left(1\right)-k \\ \Rightarrow 1=-5+3+2-k \\ \Rightarrow 1=-k \\ \Rightarrow k=-1 \end{gathered}$$

Hence, k = –1.