Question

If the remainder on dividing the polynomial:- 2x⁴-kx²+5x-3k+3 by x+2 is 4 , then find the value of K

Answer 1

$\underline{\textbf{Given:}}$

$\mathsf{The\;remainder\;when\;2x^4-kx^2+5x-3k+3\;is\;divided}$

$\mathsf{by\;x+2\;is\;4}$

$\underline{\textbf{To find:}}$

$\textsf{The value of k}$

$\underline{\textbf{Solution:}}$

$\mathsf{Since\;the\;remainder\;when\;2x^4-kx^2+5x-3k+3\;is\;divided}$

$\mathsf{by\;x+2\;is\;4}$

$\implies\mathsf{P(-2)=4}\;\;\;\;\textsf{(By Remainder theorem)}$

$\implies\mathsf{2(-2)^4-k(-2)^2+5(-2)-3k+3=4}$

$\implies\mathsf{2(16)-k(4)+5(-2)-3k+3=4}$

$\implies\mathsf{32-4k-10-3k+3=4}$

$\implies\mathsf{25-7k=4}$

$\implies\mathsf{-7k=4-25}$

$\implies\mathsf{-7k=-21}$

$\implies\mathsf{k=\dfrac{-21}{-7}}$

$\implies\boxed{\mathsf{k=3}}$

$\underline{\textbf{Remainder theorem:}}$

$\boxed{\begin{minipage}{6cm} \\\textsf{The remainder when P(x) is divided by}\\\\\textsf{(x-a) is P(a)}\\ \end{minipage}}$

Answer 2

Answer:

The value of k is 3

Step-by-step explanation:

Given,

The remainder obtained when the polynomial $2x^4-kx^2+5x-3k+3$ is divided by x + 2 is '4'.

Required to find,

The value of 'k'.

We know,

By remainder theorem, when a polynomial p(x) is divided by another polynomial q(x) = x-a, then the remainder will p(a).

Here,

P(x) =  $2x^4-kx^2+5x-3k+3$

q(x) = x+2

Then by the remainder theorem, the remainder when  $2x^4-kx^2+5x-3k+3$ is divided by x+2 is p(-2).

p(-2) = 2(-2)^4 - k(-2)^2 +5(-2) -3k +3

= 2 x 16 -k x4 -10 -3k +3

= 32 -4k -10 -3k +3

= 25 -7k

Given, P(-2) = 4

25 - 7k = 4

7k = 21

k =3

The value of k is 3