Question

Find the values of a and b so that the polynomial (x - 10x + ax + b) is
exactly divisible by (r - 1) as well as (x - 2).​

Answer 1

Question: Find the values of a and b so that the polynomial (x^3 - 10x^2 + ax + b) is  exactly divisible by (x - 1) as well as (x - 2).​

Answer:

a = 23,  b = -14

Step-by-step explanation:

Using remainder theorem, if p(x) is divided by g(x), then for root of g(x), f(x) is 0.

For g(x) = (x - 1),  x = 1:

⇒ p(1) = 0

⇒ (1)³ - 10(1)² + a(1) + b = 0

⇒ a + b = 9

For g(x) = (x - 2),  x = 2:

⇒ p(2) = 0

⇒ (2)³ - 10(2)² + a(2) + b = 0

⇒ 8 - 40 + 2a + b = 0

⇒ a + a + b = 32

⇒ a + 9 = 32             [a + b = 9]

⇒ a = 23          

   Hence, b = 9 - a = 9-23 = -14

Answer 2

Given :-

x - 10x + ax + b is  exactly divisible by (x - 1) as well as (x - 2).​

To Find :-

Find the values of a and b

Solution :-

x - 1 = 0

x = 1

Putting x as 1

0 = 1³ - 10(1)² + a(1) + b

0 = 1 - 10 + a + b

0 = -9 + a + b

9 = a + b

9 - b = a

When x - 2 = 0

x - 2 = 0

x = 2

Putting x as 2

0 = 2³ - 10(2)² + a(2) + b

0 = 8 - 10(4) + 2a + b

0 = 8 - 40 + 2a + b

0 = -32 + 2a + b

32 = 2a + b

32 = 2(9 - b) + b

32 = 18 - 2b + b

32 = 18 - b

32 - 18 = -b

-14 = -b

14 = b

Now, Finding a

a = 9 - b

a = 9 - 14

a = -5