Question

the polynomial x^4-2x^3 +3x^2_ax_b divided by x-1 and x+1leaves remainder 5 and 19 .find a and b and also find the reaminder.​

Answer 1

Answer:

Step-by-step explanation:

Given :-

f(x) = x⁴ - 2x³+ 3x² - ax + b

To Find :-

Value of a and b.

Solution :-

f(x) = x⁴ - 2x³+ 3x² - ax + b

According to Question,

When f(x) is divided by (x-1), it leaves a remainder 5

⇒ f(1) = 5

⇒ 1 - 2(1)³+ 3(1)² - a(1) + b = 5

⇒ 1 - 2 + 3 - a +b = 5

⇒ -a + b = 3 … (i)

When f(x) is divided by (x+1), it leaves a remainder 19

⇒ f(-1) = 19

⇒ (-1)⁴ - 2(-1)³ + 3(-1)² - a(-1) + b = 19

⇒ 1 + 2 + 3 + a + b = 19

⇒ a +b = 13 … (ii)

Adding (i) and (ii), we get

⇒ 2b = 16

⇒ b = 16/2

⇒ b = 8

⇒ a = b - 3

⇒ a = 8 - 3

⇒ a = 5

Hence, the value of a and b are 5 and 8.

Answer 2

Question:

The polynomial x^4 -2x^3 +3x^2 -ax -b

is divided by x-1 and x+1 to leave the remainders 5 and 19 respectively.Then find the value of a and b .

Note:

Remainder theorem:

Consider a polynomial p(x) .

If p(x) if divided by (x-a) then the reminder is given by , r = p(a).

Solution:

Let the given polynomial be;

p(x) = x^4 - 2x^3 + 3x^2 - ax - b.

Case(1);

When p(x) is divided by (x-1) , then the reminder is 5.

Thus,

As per remainder theorem, we have;

=> 5 = p(1)

=> 5 = (1)^4 -2(1)^3 +3(1)^2 - a(1) - b

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

=> 5 = 2 - a - b

=> a + b = 2 - 5

=> a + b = - 3 ----------(1)

Case(2);

When p(x) is divided by (x+1) , then the reminder is 19.

Thus,

As per remainder theorem, we have;

=> 19 = p(-1)

=> 19 = (-1)^4 -2(-1)^3 +3(-1)^2 - a(-1) - b

=> 19 = 1 + 2 + 3 + a - b

=> 19 = 6 + a - b

=> a - b = 19 - 6

=> a - b = 13

=> a = b + 13 ----------(2)

Now,

Putting a = b + 13 , in eq-(1), we get;

=> a + b = - 3

=> (b + 13) + b = - 3

=> 2b = - 3 - 13

=> 2b = - 16

=> b = -16/2

=> b = - 8

Now;

Putting b = - 8 in eq-(2) , we get;

=> a = b + 13

=> a = - 8 + 13

=> a = 5

Hence,

The required values of "a" and "b" are

5 and -8 respectively.