Question

The sum of the remainder obtained when {X power 3 + ( K + 8 )X + K } is divided by ( X - 2) or when it is divided by ( X + 1 ) is zero. Find the value of X.​

Answer 1

Question :-----

• The sum of Remainders obtained when x³+(k+8)x + k is divided by (x-2) and (x+1) we get = 0 .

To Find :----

• Value of K ?

Concept Used :------

• The remainder theorem states that when a polynomial, f(x), is divided by a linear polynomial , (x - a) , the remainder of that division will be equivalent to f(a).

Solution :------

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when x³ + (k+8)x + k is divided by (x-2) , according to remainder theoram our raminder will be :------

first x - 2 = 0 => x = 2

f(2) = (2)³ + (k+8)2 +k

→ f(2) = 8 + 2k + 16 +k

→f(2) = 3k + 24 = our remainder .

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now, again,

x+1 = 0 , => x = (-1)

so,

f(-1) = (-1)³ + (k+8)(-1) + k

→ f(-1) = -1 - k -8 + k

→ f(-1) = -9 = Required Remainder .

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Now, it has been said that , sum of both remainders is equal to zero .

so,

3k + 24 + (-9) = 0

→ 3k = 9 - 24

→ 3k = -15

→ k = -15/3 = (-5) ..

Hence , value of k is $\red{\bold{</strong><strong>k</strong><strong> = </strong><strong>(</strong><strong>-</strong><strong>5</strong><strong>)</strong><strong>}}$

(Hope it Helps you)

Answer 2

Solution :

Before solving it we have learn a theorem

Remainder theorem : If a polynomial p(x) of degree greater than or equal to 1 and a real number a and if p(x) is divided by (x - a), then p(a) is the remainder.

Now let's start solving it.

Let p(x) = x³ + (k + 8)x + k

When p(x) is divided by (x - 2)

Find the zeroes of x - 2

⇒ x - 2 = 0

⇒ x = 2

By remainder theorem

p(2) is the remainder

p(x) = x³ + (k + 8)x + k

Substitute x = 2

p(2) = 2³ + (k + 8)2 + k

= 8 + 2k + 16 + k

= 3k + 24

i.e p(2) = 3k + 24

When p(x) is divided by (x + 1)

Find the zero of x + 1

⇒ x + 1 = 0

⇒ x = - 1

By remainder theorem

p(-1) is the remainder

p(x) = x³ + (k + 8)x + k

Substitute x = - 1

p(-1) = ( - 1 )³ + (k + 8)( - 1) + k

= - 1 + k( - 1) + 8( - 1) + k

= - 1 - k - 8 + k

= - 9

i.e p(-1) = - 9

Given :

Sum of the remainder when p(x) is divided by (x - 2) and (x + 1) = 0

⇒ p(2) + p(-1) = 0

⇒ 3k + 24 + ( - 9 ) = 0

⇒ 3k + 24 - 9 = 0

⇒ 3k + 15 = 0

⇒ 3k = - 15

⇒ k = - 5

Hence, the value of k is - 5.