Question

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

Answer 1

Step-by-step explanation:

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a=-4;b=5;c=-2

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Answer 2

Given:

f(x) = x³ + ax² + bx + c.

f(1) = f(2) = 0, f(4) = f(0).

To Find:

values of a, b, and c

Solution:

According to the question,

x³ + ax² + bx + c = 0

for f(1),

(1)³ + (a×1²) + b×1 + c = 0

1 + a + b + c = 0 -----Equation(1).

for f(2),

(2)³ + (a×2²) + b×2 + c = 0

8 + 4a + 2b + c = 0 -----Equation(2).

for f(4) = f(0),

(4)³ + (a×4²) + b×4 + c = (0)³ + (a×0²) +(b×0) + c.

64 + 16a + 4b + c = 0 + 0 + 0 + c.

64 + 16a + 4b +c = c.

64 + 16 a + 4b = 0.

16 + 4a + b = 0.

b = -16 - 4a.-----Equation(3).

Putting equation (3) in equation(1),

1 + a - 16 - 4a + c = 0.

- 15 - 3a + c = 0.

c = 15 + 3a.---------Equation (4).

Putting equations (3) and (4) in equation (2),

8 + 4a - 32 - 8a + 15 + 3a = 0.

- 9 - a = 0.

a = -9.

Putting the value of a in equation (3),

b = - 16 - 4 × -9.

b = 20.

Putting the value of a in equation (4),

c = 15 + 3 × -9.

c = -12.

Hence, the values of a, b and c are -9, 20, and -12.