Question
Question: The polynomial f(x) = x<sup>4</sup> + ax<sup>3</sup> + bx<sup>2</sup> + cx + d has real coefficients...
The polynomial f(x) = x4 + ax3 + bx2 + cx + d has real coefficients and f (2i) = f(2 + i) = 0. The value of
(a + b + c + d) equals to-
A
1
B
4
C
9
D
10
Answer
9
Explanation
Solution
If a polynomial has real coefficients, then roots occur in
complex conjugate and roots are 2i, –2i, 2 + i, 2 – i.
Hence,f(x) = (x + 2i)(x – 2i)(x – 2 – i)(x – 2 + i)
f(1) = (1 + 2i)(1 – 2i)(1 – 2 – i)(1 – 2 + i)
f(1) = 5 × 2 = 10
Also, f(1) = 1 + a + b + c + d
∴ 1 + a + b + c + d = 10
⇒ a + b + c + d = 9.