You visited us 39 times! Enjoying our articles? Unlock Full Access!

Zeroes of a Polynomial

When the poly...

Question

When the polynomial $x^{4} + x^{2} + 1$ is divided by $(x + 1) (x^{2} - x + 1)$ then the remainder is $a x + b$ , then $a + b$ is equal to

1

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

2

No worries! We‘ve got your back. Try BYJU‘S free classes today!

3

No worries! We‘ve got your back. Try BYJU‘S free classes today!

4

No worries! We‘ve got your back. Try BYJU‘S free classes today!

Open in App

Solution

The correct option is A $1$
$\frac{x^{4} + x^{2} + 1}{(x + 1) (x^{2} - x + 1)}$

$= \frac{x^{4} + 2 x^{2} + 1 - x^{2}}{(x + 1) (x^{2} - x + 1)}$

$= \frac{{(x^{2} + 1)}^{2} - x^{2}}{(x + 1) (x^{2} - x + 1)}$

$= \frac{(x^{2} - x + 1) (x^{2} + x + 1)}{(x + 1) (x^{2} - x + 1)}$

$= \frac{(x^{2} + x + 1)}{(x + 1)}$

$= \frac{x (x + 1) + 1}{(x + 1)}$

$= x + \frac{1}{x + 1}$

Remainder $= 1$ is of the form $a x + b$

$\Rightarrow a = 0, b = 1$

$∴ a + b = 0 + 1 = 1$

Suggest Corrections

Similar questions

Q. When the polynomial

x^{4} + x^{2} + 1

is divided by

(x + 1) (x^{2} - x + 1)

then the reminder is

a x + b

, then

a + b

is equal to

The polynomial x⁴- 2x³+ 3x²- ax + b when divided by x +1 and x -1 give remainder 19 and 5 respectively . Find the remainder when the polynomial is divided by x -3

Q. If

f (x) = x^{4} - 2 x^{3} + 3 x^{2} - a x + b

is a polynomial such that when it is divided by

(x - 1)

and

(x + 1)

, then the remainders are

5

and

19

respectively. If

f (x)

is divided by

(x - 2)

, then the remainder is:

Q. The polynomial f(x) = x⁴ − 2x³ + 3x² − ax = b when divided by (x − 1) and (x + 1) leaves 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).

Q. If

f (x) = x^{4} - 2 x^{3} + 3 x^{2} - a x + b

is a polynomial such that when it is divided by

x - 1

and

x + 1

, the remainder are respectively

5

and

19

. Determine the remainder when

f (x)

is divided by

(x - 2)

Join BYJU'S Learning Program

When the polynomial x4+x2+1 is divided by (x+1)(x2−x+1) then the remainder is ax+b , then a+b is equal to

The correct option is A 1x4+x2+1(x+1)(x2−x+1)=x4+2x2+1−x2(x+1)(x2−x+1)=(x2+1)2−x2(x+1)(x2−x+1)=(x2−x+1)(x2+x+1)(x+1)(x2−x+1)=(x2+x+1)(x+1)=x(x+1)+1(x+1)=x+1x+1Remainder=1 is of the form ax+b⇒a=0,b=1∴a+b=0+1=1

When the polynomial $x^{4} + x^{2} + 1$ is divided by $(x + 1) (x^{2} - x + 1)$ then the remainder is $a x + b$ , then $a + b$ is equal to