Question

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

Answer 1

Answer:

The value of a is 23 and b is (-14).

Step-by-step explanation:

Polynomial = $\sf{{x}^{3} - 10 {x}^{2} + ax + b}$

Value of a and b = ?

The polynomial is supposed to divisible by (x - 1) and (x - 2)

____________________________

• Having divisor as (x - 1) :

$\longrightarrow \: \sf{x - 1 = 0} \\ \longrightarrow \:\: \: \: \: \: \: \: \:\sf{x = 1}$

Substitute the value of x in the given polynomial:

$\longrightarrow \: \sf{{x}^{3} - 10 {x}^{2} + ax + b}$

$\longrightarrow\sf{1}^{3}- {(10 \times 1}^{2}) +(a \times 1) + b = 0$

$\longrightarrow{\sf{1 - 10 + a + b = 0}}$

$\sf{\longrightarrow{- 9 + a + b = 0}}$

$\sf{\longrightarrow{ a + b = 9 \: \gray{ - - - (1)}}}$

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• Having (x - 2) as divisor :

$\longrightarrow \: \sf{x - 2= 0} \\ \longrightarrow \:\: \: \: \: \: \: \: \:\sf{x = 2}$

Substitute the value of x in the given polynomial:

$\longrightarrow{\sf{{x}^{3} - 10 {x}^{2} + ax + b}}$

$\sf{\longrightarrow{{(2)}^{3} - (10 \times {2}^{2}) + (a \times 2) + b = 0}}$

$\sf{\longrightarrow} \: 8 - 40 + 2a + b = 0$

$\sf{\longrightarrow} \: - 32 + 2a + b = 0$

$\sf{\longrightarrow} \: 2a + b = 32$

$\sf{\longrightarrow} \: b = 32 - 2a \: \: \gray{ - - - (2)}$

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• Having equation (1) and (2) :

Substitute the value of b in equation (1) -

$\sf{\longrightarrow}\:a + (32 - 2a) = 9$

$\sf{\longrightarrow}\:32 - a = 9$

$\sf{\longrightarrow}\:-a = 9 - 32$

$\sf{\longrightarrow}\:a = 23$

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• Value of b :

$\sf{\longrightarrow}\:b = 32 - 2a$

$\sf{\longrightarrow}\:b = 32 - 2(23)$

$\sf{\longrightarrow}\:b = 32 - 46$

$\sf{\longrightarrow}\:b = -14$

Therefore, the value of a is 23 and b is (-14).

Answer 2

$\huge \leadsto \frak {\red {\boxed{\frak {23 \: and \: -14}}}}$

The given polynomial is divisible by

(x -1) (x-2)

According to the question,

x - 1 = 0

x = 0 + 1

x = 1

Putting the value of x

1³ - (10 × 1²) + a×1+ b = 0

1 - 10 + a + b = 0

-9 + a + b = 0

a + b = 9 (Eqⁿ 1)

Eqⁿ 1 and 2

(2)³ -(10 × 2²) + (a × 2) + b = 0

8 - 40 + 2a + b = 0

-32 + 2a + b = 0

b = 32 - 2a

Substitute the value of b

a (32 - 2a) = 9

32-(2a - a) = 9

32 - a = 9

-a = 9 - 32

a = 23

Now

Finding b

b = 32 - 2(23)

b = 32 - 46

b = -14