Answer: The required remainder is 5.
Step-by-step explanation: We are given to find the remainder when we divide the following cubic polynomial by (x+2) :
[tex]f(x)=2x^3+4x^2-x+3~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(i)[/tex]
We have the following theorem :
Remainder theorem : If a polynomial p(x) is divided by the factor (x-a), then the remainder is p(a).
So, for the given linear factor, we have
[tex]x+2=0\\\\\Rightarrow x=-2.[/tex]
Substituting x = -2 in equation (i), we get
[tex]f(-2)\\\\=2(-2)^3+4(-2)^2-(-2)+3\\\\=-16+16+2+3\\\\=5.[/tex]
Thus, the required remainder is 5.
