Question

If x³ + ax² − bx+ 10 is divisible by x² − 3x + 2, find the values of a and b.

Answer 1

Let f(x) = x³ + ax² − bx + 10 and g(x) = x² − 3x + 2 be the given polynomials. 

We have g(x) = x² − 3x + 2 = (x − 2) (x − 1)

Clearly, (x − 1) and (x − 2) are factors of g(x)

Given that f(x) is divisible by g(x)

g(x) is a factor of f(x)

(x − 2) and (x − 1) are factors of f(x)

From factor theorem

f(x − 1) and (x − 2) are factors of f(x) then f(1) = 0 and f(2) = 0 respectively.

f(1) = 0

(1)³ + a(1)² − b(1) + 10 = 0

1 + a − b + 10 = 0

a − b + 11 = 0     ...(i)

f(2) = 0

(2)³ + a(2)² − b(2) + 10 = 0

8 + 4a − 2b + 10 = 0

4a − 2b + 18 = 0

2(2a − b + 9) = 0

2a − b + 9 = 0    ...(ii)

Subtract (i) from (ii), we get

2a − b + 9 −(a − b + 11) = 0

2a − b + 9 − a + b − 11 = 0

a − 2 = 0

Putting value of a in (i), we get

2 − b + 11 = 0

b = 13

Hence,

a = 2 and b = 13