Question

The sum of the remainders obtained when (x³ + (k + 8)x + k) is divided by (x - 2) or when it is divided by (x + 1) is zero. Find the value of k. (no spam❗) ​

Answer 1

Given :-

• P(x) = (x³ + (k + 8)x + k)

• f (x) = ( x - 2)

• f(x) = (x + 1)

• The sum of the remainder obtained is zero.

To Find:-

• The Value of K

Now,

→ f(x) = x - 2 = 0

→ x = 2

Putting the value of x in P(x)

→ P(x) = x³ ( k + 8 )x + k

→ P(2) = (2)³ + ( k + 8)2 + k

→ 8 + 2k + 16 + k

→ 24 + 3k....eq. 1

Now, again

→ x + 1 = 0

→ x = -1

Putting the value of x in P(x)

→ P(x) = x³ + (k + 8)x + k

→ (-1)³ + ( k + 8)-1 + k

→ -1 - k - 8 + k

→ -9.... eq. 2

Adding eq 1 and eq 2

→ 24 + 3k +(-9) = 0

→ 24 - 9 + 3k = 0

→ 15 + 3k = 0

→ 3k = -15

→ k = -15/3

→ k = -5

Hence, The Value of k is -5.

Answer 2

$\huge{\boxed{\rm{\red{Question}}}}$

The sum of the remainders obtained when (x³ + (k + 8) x + k) is divided by (x - 2) or when it is divided by (x + 1) is zero. Find the value of k.

$\huge{\boxed{\rm{\red{Answer}}}}$

$\large{\boxed{\boxed{\sf{Given \: that}}}}$

• (x³ + (k + 8)x + k).

• (x - 2).

• The sum of the remainders obtained is zero (0).

• (x + 1).

$\large{\boxed{\boxed{\sf{To \: find}}}}$

• Value of k.

$\large{\boxed{\boxed{\sf{Assumptions}}}}$

• Let's x³ + (k + 8)x + k) is the value of p(x).

• Let's (x - 2) is the value of f(x).

• Let's (x - 2) is the value of f(x) also.

$\large{\boxed{\boxed{\sf{Solution}}}}$

• Value of k = -5

$\large{\boxed{\boxed{\sf{What \: the \: question \: says \: ?}}}}$

$\large{\boxed{\boxed{\sf{Let's \: understand \: the \: concept \: 1^{st}}}}}$

• The question state that There is a sum of the remainder when it is divided by ( x³ + ( k + 8 ) x + k is divided by ( x - 2 ) after that's ( or ) when it I divided by ( x + 1 ) then the result come is 0. Then the question ask that we have to find the value of k.

$\large{\boxed{\boxed{\sf{How \: to \: solve \: this \: question \: ?}}}}$

$\large{\boxed{\boxed{\sf{See \: the \: procedure \: that \: is \: given \: below \: !}}}}$

• Firstly we have to find the value of x Then we have to put the value of x in P(x) = x³ ( k + 8 )x + k. Then we have to agin find the value of x. After that we have to put the value of x in P(x) = x³ ( k + 8 )x + k. Now we have to add equation 1 with equation 2. Then we get our Answer.

$\large{\boxed{\boxed{\sf{Full \: solution}}}}$

$\large\orange{\texttt{According to the question,}}$

Finding the value of x –

$\mapsto$ f(x) = x - 2 = 0

$\mapsto$ f(x) = x = 0 + 2

$\mapsto$ f(x) = x = 2

$\bold{\green{\fbox{\green{Value of x is 2}}}}$

$\large\purple{\texttt{Putting the value of x in P(x) }}$

$\leadsto$ P(x) = x³ ( k + 8 )x + k

$\leadsto$ P(2) = (2)³ + ( k + 8 )2 + k

$\leadsto$ 8 + 2k + 16 + k

$\leadsto$ = 8 + 16 + 2k + k

$\leadsto$ = 24 + 3k $\large\mathtt\red{Equation \: 1}$

Finding the value of x again –

$\mapsto$ x + 1 = 0

$\mapsto$ x = 0 - 1

$\mapsto$ x = -1

$\bold{\green{\fbox{\green{Value of x is -1}}}}$

$\large\purple{\texttt{Putting the value of x in P(x) }}$

$\leadsto$ P(x) = x³ ( k + 8 )x + k

$\leadsto$ -1³ + ( k + 8 )-1 + k

$\leadsto$ -1 -k -8 + k

$\leadsto$ -1 - 8 -k +k

$\leadsto$ -9 $\large\mathtt\red{Equation \: 2}$

Adding equation 1 and 2 –

$\mapsto$ 24 + 3k + (-9) = 0

$\mapsto$ 24 - 9 + 3k = 0

$\mapsto$ 15 + 3k = 0

$\mapsto$ 3k = 0 - 15

$\mapsto$ 3k = -15

$\mapsto$ k = -15/3

$\mapsto$ k = -5

$\huge{\underline{\mathtt{\blue{A}\red{N}\pink{S}\green{W}\purple{E}\orange{R}}}}$ 5 is the value of k

Hope it's helpful

Thank you :)