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

Using Factor Theorem
According to the Factor Theorem, if \((x - a)\) is a factor of a polynomial \(f(x)\), then \(f(a) = 0\).

Here, the polynomial is
\(f(x) = x^3 - kx^2 + 15x - 6\)
Since \((x - 3)\) is a factor, we substitute \(x = 3\).

Substituting \(x = 3\)
\(f(3) = (3)^3 - k(3)^2 + 15(3) - 6\)
\(= 27 - 9k + 45 - 6\)
\(= 66 - 9k\)

Applying the Factor Theorem
Since \(f(3) = 0\),
\(66 - 9k = 0\)
\(9k = 66\)
\(k = \frac{66}{9}\)
\(k = \frac{22}{3}\)

Final Answer
The value of \(k\) is \(\frac{22}{3}\).