Question

find the value of 'k' if 4x^3 - 2x^2 + kx + 5 leaves remainder -10 when divided by 2x + 1

Answer 1

Hope this will help u

Answer 2

Answer:

Answer:

$\sf \: The \: value \: of \: k \: is \: \bold{ 28.}$

Step-by-step explanation:

We have the following theorem :

Remainder Theorem : When a polynomial p(x) is divided by a linear factor (x-a), then the remainder is p(a).

The given polynomial

$\begin{gathered} \sf \: p(x)=4x^3-2x^2+kx+5p(x)=4x +kx+5 \\ \\ \sf \: leaves \: remainder \: -10, \: when \: divided \: by \: 2x+1.\end{gathered}$

Now,

$\begin{gathered} \sf \: 2x+1=0\\\\\Rightarrow \sf \: x=-\dfrac{1}{2}.\end{gathered}$

Therefore, we must have

$\begin{gathered}\begin{gathered} \sf \: p(\frac{1}{2})=-10\\\\ \sf \implies \sf \: 4\times(-\frac{1}{2})^3-2\times(-\frac{1}{2})^2+k\times(-\frac{1}{2})+5=-10\\\\\\\sf \implies-\dfrac{1}{2}-\dfrac{1}{2}-\dfrac{k}{2}+5=-10\\ \\ \\ \sf \implies \sf \: -\dfrac{k}{2}+4=-10\\ \\ \sf \implies \sf-\dfrac{k}{2}=--14\\ \\ \sf \implies \sf \boxed{ \red{ \sf k=28}}.\end{gathered}\end{gathered}$

Thus, the required value of k is 28