Question

The sum of the remainder 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.​

Answer 1

$\huge\bf{Answer:-}$

Refer the attachment (1) & (2)

Answer 2

$\color {red}\huge\bold\star\underline\mathcal{Question:-}$ we have to Find value of k ?

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$\huge\underline\mathfrak\green{Remember}$ :--- If p(x) is a polynomial of degree n which is greater than or equal to one and a is any real number which will be the divisor x-a would be the factor of the polynomial if the remainder is 0.

It is given in question that (x-2) & (x+1) gives remainder 0 . so they are factors of (x³+(k+8)x+k) .

Now,

f(2) = (2)³+ (k + 8)×2+ k

f(2) = 3k + 24

$\large\red{\boxed{\sf </strong><strong>Similarly</strong><strong>}}$

f( - 1) = (- 1)³ + (k + 8)( - 1) + k

f(-1) = (-9)

since both are giving remainder 0,

they are equal ..

so,

$3k + 24 = ( - 9) \\ \\ 3k = - 9 - 24 \\ \\ k = \frac{ - 33}{3} = ( - 11) \: (ans.)$

$\huge\underline\mathfrak\green{Hope\:it\:Helps\:You}$