(x^{2} – 1) is a factor of f(x) = (x^{5} + ax^{4} + bx^{3} + cx^{2} + x + d). The graph of f(x) intersects the Y axis at (0, –3). Find the value of (a + c).

Options:

A . 1

B . 4

C . 2

D . 5

E . 3

Answer: Option E : E (x^{2} -1) = (x – 1) (x + 1) is a factor of f(x). So, f(1) = f(-1) = 0 1 + a + b + c + 1 + d = –1 + a + b + c – 1 + d b = -2 (1 + a + b + c + 1 + d) = 0 (–1 + a + b + c – 1 + d) = 0 i.e., a + b + c + d = 2 a – b + c + d = 2. Since graph passes through (0, –3) f(0) = –3 or d = –3 ∴a + b + c = 1 and a – b + c = 5 ∴a + c = 3 Hence option (e)

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