For the expression f(x) = x3 + ax2 + bx + c if f(1) = f(2) = 0 and f(4

1.	For the expression f(x) = x3 + ax2 + bx + c if f(1) = f(2) = 0 and f(4) = f(0). Find the values of a, b, c.A) a = -8, b = -20, c = 12B) a = 9, b = -20, c = -12C) a = -9, b = 20, c = -12D) a = -8, b =. 20, c = -12
Answer» Correct option is (C) a = -9, b = 20, c = -12 \(\because\) f(1) = 0 \(\Rightarrow\) \(1^3+a.1^2+b.1+c=0\) \(\Rightarrow\) a+b+c = -1 ________(1) \(\because\) f(2) = 0 \(\Rightarrow\) \(2^3+a.2^2+b.2+c=0\) \(\Rightarrow\) 4a+2b+c = -8 ________(2) \(\because\) f(4) = f(0) \(\Rightarrow\) \(4^3+a.4^2+b.4+c=c\) \(\Rightarrow\) 64+16a+4b = 0 \(\Rightarrow\) 16a+4b = -64 \(\Rightarrow\) 4a+b = -16 \(\Rightarrow\) b = -16 - 4a ________(3) By putting b = -16 - 4a from (3) into (1) and (2), we get a - 16 - 4a + c = -1 \(\Rightarrow\) 3a - c = -15 ________(4) and 4a - 32 - 8a + c = -8 \(\Rightarrow\) 4a - c = -24 ________(5) Subtract (4) from (5), we get (4a - c) - (3a - c) = -24 - (-15) \(\Rightarrow\) a = -24+15 = -9 Put a = -9 in equation (4), we get 3 \(\times\) -9 - c = -15 \(\Rightarrow\) c = -27+15 = -12 Put a = -9 in equation (3), we get b = -16 - 4 \(\times\) -9 = -16+36 = 20 \(\therefore\) a = -9, b = 20, c = -12 Correct option is C) a = -9, b = 20, c = -12

For the expression f(x) = x3 + ax2 + bx + c if f(1) = f(2) = 0 and f(4) = f(0). Find the values of a, b, c.A) a = -8, b = -20, c = 12B) a = 9, b = -20, c = -12C) a = -9, b = 20, c = -12D) a = -8, b =. 20, c = -12

Answer»

Correct option is (C) a = -9, b = 20, c = -12

\(\because\) f(1) = 0

\(\Rightarrow\) \(1^3+a.1^2+b.1+c=0\)

\(\Rightarrow\) a+b+c = -1 ________(1)

\(\because\) f(2) = 0

\(\Rightarrow\) \(2^3+a.2^2+b.2+c=0\)

\(\Rightarrow\) 4a+2b+c = -8 ________(2)

\(\because\) f(4) = f(0)

\(\Rightarrow\) \(4^3+a.4^2+b.4+c=c\)

\(\Rightarrow\) 64+16a+4b = 0

\(\Rightarrow\) 16a+4b = -64

\(\Rightarrow\) 4a+b = -16

\(\Rightarrow\) b = -16 - 4a ________(3)

By putting b = -16 - 4a from (3) into (1) and (2), we get

a - 16 - 4a + c = -1

\(\Rightarrow\) 3a - c = -15 ________(4)

and 4a - 32 - 8a + c = -8

\(\Rightarrow\) 4a - c = -24 ________(5)

Subtract (4) from (5), we get

(4a - c) - (3a - c) = -24 - (-15)

\(\Rightarrow\) a = -24+15 = -9

Put a = -9 in equation (4), we get

3 \(\times\) -9 - c = -15

\(\Rightarrow\) c = -27+15 = -12

Put a = -9 in equation (3), we get

b = -16 - 4 \(\times\) -9 = -16+36 = 20

\(\therefore\) a = -9, b = 20, c = -12

Correct option is C) a = -9, b = 20, c = -12