Question

If (x – 2) is a factor of \[(x^{3} + 2x^{2} - kx + 10)\], find the value of k. Hence, determine whether (x + 5) is also a factor of the given expression. 

[Hint : Let \(f(x) = x^{3} + 2x^{2} - kx + 10\). Then, f(2) = 0, gives k = 13.
Thus, \(f(x) = x^{3} + 2x^{2} - 13x + 10\). Now, show that f(–5) = 0.]

Answer 1

Let \(f(x) = x^{3} + 2x^{2} - kx + 10\).

Since \((x - 2)\) is a factor, by Factor Theorem, \(f(2) = 0\). 

Now, \(f(2) = 2^{3} + 2(2)^{2} - k(2) + 10\)

= \(8 + 8 - 2k + 10\)

= \(26 - 2k\) 

So, \(26 - 2k = 0 \Rightarrow k = 13\).

Thus, \(f(x) = x^{3} + 2x^{2} - 13x + 10\).

Now check whether \((x + 5)\) is a factor:

\(f(-5) = (-5)^{3} + 2(-5)^{2} - 13(-5) + 10\)

= \(-125 + 50 + 65 + 10\)

= \(0\) 

Since \(f(-5) = 0\), \((x + 5)\) is also a factor of the given expression.