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Byju's Answer

Standard IX

Mathematics

Synthetic Division of Polynomials

Let rx be t...

Question

Let $r (x)$ be the remainder when the polynomial $x^{135} + x^{125} - x^{115} + x^{5} + 1$ is divided by $x^{3} - x$ . Then:

r (x)

is the zero polynomial

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r (x)

is a nonzero constant

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degree of

r (x)

is one

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degree of

r (x)

is two

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Solution

The correct option is C degree of $r (x)$ is one
It is obvious that if we divide by $x^{3} - x$ , then remainder will always be degree less than $1$ .

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Similar questions

Q. Let

r (x)

be the remainder when the polynomial

x^{135} + x^{125} - x^{115} + x^{5} + 1

is divided by

x^{3} - x

. Then

On dividing a polynomial p(x) by a nonzero polynomial q(x), let g(x) be the quotient and r(x) be the remainder then p(x) = q(x).g(x)+r(x), where

(a) r(x) = 0 always

(b) deg r(x) < deg g(x) always

(d) r(x) = g(x)

Consider an unknown polynomial which when divided by

(x - 3)

and

(x - 4)

leaves remainders

2

and

1

respectively. Let

R (x)

be the remainder when this polynomial is divided by

(x - 3) (x - 4)

Range of

f (x) = \frac{R (x)}{(x^{2} - 3 x + 2)}

Q. The remainder on dividing the polynomial

q (x)

(x - a)

k

and the remainder on dividing the polynomial

r (x)

(x - a)

- k

.
(a) Find

q (a)

.
(b) Prove that

(x - a)

is a factor of the polynomial

q (x) + r (x)

Q. Consider an unknown polynomial which when divided by

(x - 3)

and

(x - 4)

leaves remainders

2

and

1

respectively. Let

R (x)

be the remainder when this polynomial is divided by

(x - 3) (x - 4)

. If

R (x) = p x^{2} + (q - 1) x + 6

has no distinct real roots and

p > 0

, then the value of

3 p + q

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Let r(x) be the remainder when the polynomial x135+x125−x115+x5+1 is divided by x3−x. Then:

The correct option is C degree of r(x) is oneIt is obvious that if we divide by x3−x, then remainder will always be degree less than 1.

Let $r (x)$ be the remainder when the polynomial $x^{135} + x^{125} - x^{115} + x^{5} + 1$ is divided by $x^{3} - x$ . Then:

The correct option is C degree of $r (x)$ is one
It is obvious that if we divide by $x^{3} - x$ , then remainder will always be degree less than $1$ .