Question

Find the value of $k$ if $(x - 3)$ is a factor of the polynomial $x^3 - kx^2 + 15x - 6$.

Hint:

If $(x - a)$ is a factor, directly substitute $x = a$ using the Factor Theorem.

Answer 1

Concept: If $(x - a)$ is a factor of a polynomial $f(x)$, then by the Factor Theorem: \[ f(a) = 0 \]
Step 1: Apply Factor Theorem. Given polynomial: \[ f(x) = x^3 - kx^2 + 15x - 6 \] Since $(x - 3)$ is a factor: \[ f(3) = 0 \]
Step 2: Substitute $x = 3$. \[ 3^3 - k(3^2) + 15(3) - 6 = 0 \] \[ 27 - 9k + 45 - 6 = 0 \]
Step 3: Simplify. \[ (27 + 45 - 6) - 9k = 0 \] \[ 66 - 9k = 0 \]
Step 4: Solve for $k$. \[ 9k = 66 \] \[ k = \frac{66}{9} = \frac{22}{3} \]
Correction Check: Re-evaluating arithmetic: \[ 27 + 45 = 72,\quad 72 - 6 = 66 \] So calculation is correct.
But wait: Factor must give integer root consistency. Re-check substitution carefully: \[ f(3) = 27 - 9k + 45 - 6 = 66 - 9k \] Setting $=0$: \[ 9k = 66 \Rightarrow k = \frac{22}{3} \]
Conclusion: The value of $k$ is $\frac{22}{3}$.