if x3 + ax2 - bx + 10 is divisible by x2 - 3x + 2, find the values

1.	if x3 + ax2 - bx + 10 is divisible by x2 - 3x + 2, find the values of a and b.
Answer» 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 a = 2 Putting value of a in (i), we get 2 – b + 11 = 0 b = 13 Hence, a = 2 and b = 13

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

Answer»

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

a = 2

Putting value of a in (i), we get

2 – b + 11 = 0

b = 13

Hence,

a = 2 and b = 13