Given:
[tex]f(x)=x^3+4x^2+5x+2,k=-2[/tex]
To find:
The the given function in the form [tex]f(x)=(x-k)q(x)+r[/tex].
Solution:
We have,
[tex]f(x)=x^3+4x^2+5x+2[/tex]
The coefficients of f(x) are 1, 4, 5, 2.
Divide the given function by (x+2) by using synthetic division as shown below:
-2 | 1 4 5 2
-2 -4 -2
_____________________
1 2 1 0
_____________________
Bottom row represent the quotient and last element of the bottom row is the remainder.
Degree of the function is 3 and the degree of division is 1. So, the degree of the quotient is 3-1 = 2.
So, the quotient is [tex]x^2+2x+1[/tex] and the remainder is 0.
Now,
[tex]f(x)=(x-k)q(x)+r[/tex]
Here, q(x) is the quotient and r is the remainder.
[tex]f(x)=(x-k)(x^2+2x+1)+0[/tex]
Therefore, the required answer is [tex]f(x)=(x-k)(x^2+2x+1)+0[/tex].
