the-expression-ax-2-bx-c-is-divisible-by-x-1-has-reminder-2-when-divided-by-x-1-and-has-reminder-8-when-divided-by-x-2-find-the-value-of-a-b-and-c-

Question Number 9575 by j.masanja06@gmail.com last updated on 18/Dec/16

the expression {ax}^{2} + bx + c is divisible by x - 1, has reminder 2 when divided by x + 1, and has reminder 8 when divided by x - 2 . find the value of a, b and c .

Commented by ridwan balatif last updated on 18/Dec/16

P (x) \equiv {ax}^{2} + bx + c

(x - 1) is divisible P (x) \to P (1) = 0

a {(1)}^{2} + b (1) + c = 0

a + b + c = 0 \dots (1)

when P (x) is divided by (x + 1), P (x) has reminder 2, {it}^{'} s means P (- 1) = 2

a {(- 1)}^{2} + b (- 1) + c = 2

a - b + c = 2 \dots (2)

when P (x) is divided by (x - 2), P (x) has reminder 8, {it}^{'} s means P (2) = 8

a {(2)}^{2} + b (2) + c = 8

4 a + 2 b + c = 8 \dots (3)

(1) - (2)

a + b + c = 0

a - b + c = 2

- - - - - - (-)

2 b = - 2

b = - 1 \dots *

(1) - (3)

a + b + c = 0

4 a + 2 b + c = 8

- - - - - - - (-)

- 3 a - b = - 8

- 3 a - (- 1) = - 8

- 3 a = - 7

a = \frac{7}{3} \dots * *

a + b + c = 0

\frac{7}{3} + (- 1) + c = 0

c = - \frac{4}{3} \dots * * *

so, a = \frac{7}{3}, b = - 1, c = - \frac{4}{3}

P (x) \equiv \frac{7}{3} x^{2} - x + \frac{4}{3}