Question

Find the value of k for which (x + 3) is a factor of x4 – x3 – 11x2 – x + k.​

Answer 1

Value of k = -12

Given:

(x+3) is a factor of the polynomial $x^{4} - x^{3} - 11x^{2} - x + k$

To find:

Value of k

Solution:

If x+3 is factor we can find value of x by,

=> x + 3 = 0

=> x = -3

Given the polynomial, $x^{4} - x^{3} - 11x^{2} - x + k$ we can substitute the value of x as -3, equate the polynomial to 0, and find the value of k.

$x^{4} - x^{3} - 11x^{2} - x + k$ = 0

Putting x = -3

=> $-3^{4} - (-3^{3}) - 11(-3^{2} ) - (-3) + k = 0$

=> 81 - (-27) - 11(9) - (-3) + k = 0

=> 81 + 27 - 99 + 3 + k = 0

=> 111 - 99 + k =0

=> 12 + k = 0

∴ k = -12

Hence when k = -12, (x+3) is a factor of the given polynomial

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Answer 2

Given:

(x+3) is a factor of  $x^{4}$ – $x^{3}$ – 11$x^{2}$ – x + k

To find:

Value of k

Solution:

If x+3 is factor then,

x + 3 = 0

x = -3

Now, substitute x = -3 and equate the polynomial to 0 to find the value of k.

Putting x = -3,

81 - (-27) - 11(9) - (-3) + k = 0

81 + 27 - 99 + 3 + k = 0

111 - 99 + k =0

12 + k = 0

-12 = k

Hence, k = -12