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Byju's Answer

Standard XII

Mathematics

Binomial Theorem

Let p x =x4...

Question

Let $p (x) = x^{4} + 2 x^{3} - 2 x^{2} + x - 1$
$q (x) = x^{2} + 2 x - 3$ then p(x) is divisible by q(x) if we

add

(x - 2)

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add

(x - 3)

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add

(2 - x)

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add

(3 - x)

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Solution

The correct option is A add $(x - 2)$
Lets divide $p (x) = x^{4} + 2 x^{3} - 2 x^{2} + x - 1$ by $q (x) = x^{2} + 2 x - 3$ as shown above:

Hence $- x + 2$ should be subtracted that is $(x - 2)$ should be added to $p (x) = x^{4} + 2 x^{3} - 2 x^{2} + x - 1$ so that the resulting polynomial is exactly divisible by $q (x) = x^{2} + 2 x - 3$ .

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Similar questions

Now for each pair of polynomials p(x) and q(x) given below, find p(x) + q(x) and p(x) − q(x). Also compute p(1) + q(1) and p(1) − q(1) for each.

(i)

(ii)

(iii)

(iv)

(v)

Q. By using factor theorem in the following examples, determine whether q ( x ) is a factor p ( x ) or not.

(i)

p (x) = x^{3} - x^{2} - x - 1, q (x) = x - 1

(ii)

p (x) = 2 x^{3} - x^{2} - 45, q (x) = x - 3

Now in each of the following, find the quotient and remainder on dividing p(x) by q(x) and write then in the form p(x) = u(x) q(x) + v(x).

(i)

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

Q. A polynomial

p (x)

is divided by

g (x)

, the obtained quotient

q (x)

and the remainder

r (x)

are given in the table. Find

p (x)

in each case.

Sl.	$p (x)$	$g (x)$	$q (x)$	$r (x)$
i	?	$x - 2$	$x^{2} - x + 1$	$4$
ii	?	$x + 3$	$2 x^{2} + x + 5$	$3 x + 1$
iii	?	$2 x + 1$	$x^{3} + 3 x^{2} - x + 1$	$0$
iv	?	$x - 1$	$x^{3} - x^{2} - x - 1$	$2 x - 4$
v	?	$x^{2} + 2 x + 1$	$x^{4} - 2 x^{2} + 5 x - 7$	$4 x + 12$

Q. Let

P (x)

and

Q (x)

be two polynomials. Suppose that

f (x) = P (x^{3}) + x Q (x^{3})

is divisible by

x^{2} + x + 1

, then

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Let p(x)=x4+2x3−2x2+x−1q(x)=x2+2x−3 then p(x) is divisible by q(x) if we

The correct option is A add(x−2)Lets divide p(x)=x4+2x3−2x2+x−1 by q(x)=x2+2x–3 as shown above:Hence −x+2 should be subtracted that is (x−2) should be added to p(x)=x4+2x3−2x2+x−1 so that the resulting polynomial is exactly divisible by q(x)=x2+2x–3.

Let $p (x) = x^{4} + 2 x^{3} - 2 x^{2} + x - 1$
$q (x) = x^{2} + 2 x - 3$ then p(x) is divisible by q(x) if we