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The values of a, b and c respectively for the expression f(x) = x3 + ax2 + bx + c, if f(1) = f(2) = 0 and f(4) = f(0) are :

(a) 9, 20, 12 

(b) –9, –20, 12 

(c) –9, 20, –12 

(d) –9, –20, –12

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Best answer

(c) –9, 20, –12 

Given, f(x) = x3 + lx2 + mx + n. 

f(1) = f(2) = 0 ⇒ (x – 1) and (x – 2) are factors of f(x). 

Since, f(x) is polynomial of degree 3, it shall have three linear factors. So, let the third factor be (x – k). 

Then, f(x) = (x – 1) (x – 2) (x – k) 

⇒ f(x) = x3 + lx2 + mx + n = (x – 1) (x – 2) (x – k) 

Given, f(4) = f(0) 

⇒ (4 – 1) (4 – 2) (4 – k) = (–1) (–2) (–k) 

⇒ 24 – 6k = – 2k ⇒ 4k = 24 ⇒ k = 6

∴ f(x) = (x – 1) (x – 2) (x – 6) = (x2 – 3x + 2) (x – 6) 

= x3 – 9x2 + 20x – 12

∴ x3 + lx2 + mx + n = x3 – 9x2 + 20x – 12 

⇒ l = – 9, m = 20, n = – 12.

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