For what value of k, will the expression (3x3 – kx2 + 4x + 16) be di

1.	For what value of k, will the expression (3x3 – kx2 + 4x + 16) be divisible by (x –\(\frac{k}{2}\)) ?(a) 4 (b) –4 (c) 2 (d) 0
Answer» (b) – 4. Let f(x) = 3x³ – kx² + 4x + 16. Then, f(x) will be divisible by \(\big(\)x – \(\frac{k}{2}\)\(\big)\) if f\(\big(\frac{k}{2}\big)\) = 0 ⇒ 3.\(\big(\frac{k}{2}\big)\)³ – k.\(\big(\frac{k}{2}\big)\)² + 4\(\big(\frac{k}{2}\big)\) + 16 = 0 ⇒ \(\frac{3k^3}{8}\) - \(\frac{k^3}{4}\) + \(\frac{4k}{2}\) + 16 = 0 ⇒ \(\frac{3k^3-2k^3+16k+128}{8}\) = 0 ⇒ k³ + 16k + 128 = 0 ⇒ (k + 4) (k² – 4k + 32) = 0 ⇒ k + 4 = 0 ⇒ k = – 4.

For what value of k, will the expression (3x3 – kx2 + 4x + 16) be divisible by (x –\(\frac{k}{2}\)) ?(a) 4 (b) –4 (c) 2 (d) 0

Answer»

(b) – 4.

Let f(x) = 3x³ – kx² + 4x + 16. Then, f(x) will be divisible by \(\big(\)x – \(\frac{k}{2}\)\(\big)\) if f\(\big(\frac{k}{2}\big)\) = 0

⇒ 3.\(\big(\frac{k}{2}\big)\)³ – k.\(\big(\frac{k}{2}\big)\)² + 4\(\big(\frac{k}{2}\big)\) + 16 = 0

⇒ \(\frac{3k^3}{8}\) - \(\frac{k^3}{4}\) + \(\frac{4k}{2}\) + 16 = 0

⇒ \(\frac{3k^3-2k^3+16k+128}{8}\) = 0

⇒ k³ + 16k + 128 = 0 ⇒ (k + 4) (k² – 4k + 32) = 0

⇒ k + 4 = 0 ⇒ k = – 4.