Question

The polynomial f (x)=x^4 - 2x^3 + 3x^2 -ax + b when divided by (x-1) and (x+1) leave the remainder 5 and 19 respectively. Find the values of a and b . Hence find the remainder when f (x) is divided by (x-2).

Answer 1

after following this you will get your answer

Answer 2

Given :

• A polynomial f(x) = x⁴ - 2x³ + 3x² - ax + b leaves remainder 5 when divided with (x-1) and leaves remainder 19 when divided with (x-2).

To Find :

• The values of a & b
• The remainder when f(x) is divided by (x-2)

Let us take the divisor (x-1)

( x - 1 ) = 0

$\red{x = 1}$

Substituting the value of x in f(x),

⟹ (1)⁴ - 2(1)³ + 3(1)² - a(1) + b = 5

Here, we made f(x) = 5 because it is given that when divided, it leaves remainder as 5.

$1 - 2 + 3 - a + b = 5$

$- a + b = 5 - 2$

b = 3 + a _____ (1)

Now, Let's solve for f(x) when it is divided by (x+1) leaving remainder 19.

( x + 1 ) = 0

$\blue{x = - 1}$

Substituting the value of x in f(x),

(-1)⁴ - 2(-1)³ + 3(-1)² - a(-1) + b

$⟹\:1+2+3+a+b$

From equation (1), let us substitute the value of b,

$⟹\:1+2+3+a+3+a$

$⟹\:9 + 2a = 12$

$⟹\:2a = 19 - 9$

$\boxed{a = 5}$

We got the value of 5, let us substitute this in equation (1)

⟹ b = 3 + 5

$\boxed{\:b=8}$

We found the values of both a and b.

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By trial & error method let's find the remainder when f(x) is divided by ( x - 2 )

⟹ x - 2 = 0

$\blue{x=2}$

f(x) = x⁴ - 2x³ + 3x² - 5x + 8

f(2) = (2)⁴ - 2(2)³ + 3(2)² - 5(2) + 8

= 16 - 16 + 12 - 10 + 8

= 2 + 8

= 10

Hence, the remainder we get when f(x) is divided by ( x - 2 ) is 10.

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