Question

For what value of k, is the polynomial f(x) = 3x⁴ – 9x³ + x² + 15x + k completely divisible by 3x² – 5?

Answer 1

Let f(x) = 3x⁴ – 9x³ + x² + 15x + k

It is given that f(x) is completely divisible by 3x² – 5.

Therefore, one factor of f(x) is (3x² – 5).

We get another factor of f(x) by dividing it with (3x² – 5).

On division, we get the quotient x² – 3x + 2 and the remainder k + 10.

Since, f(x) = 3x⁴ – 9x³ + x² + 15x + k completely divisible by 3x² – 5

Therefore, remainder must be zero.

$$\begin{gathered} \Rightarrow k+10=0 \\ \Rightarrow k=-10 \end{gathered}$$

Hence, the value of k is –10.