To determine the value of [tex]\( k \)[/tex] such that the remainder is zero when dividing the polynomial [tex]\( x^3 - kx^2 + 2x - 4 \)[/tex] by [tex]\( x - 2 \)[/tex], we follow these steps:
1. Set up the division:
We are given the polynomial [tex]\( P(x) = x^3 - kx^2 + 2x - 4 \)[/tex].
We need to divide this by [tex]\( (x - 2) \)[/tex].
2. Perform polynomial division:
The goal is to express [tex]\( P(x) \)[/tex] in the form:
[tex]\[
P(x) = (x - 2)Q(x) + R
\][/tex]
where [tex]\( Q(x) \)[/tex] is the quotient and [tex]\( R \)[/tex] is the remainder.
3. Use the division algorithm:
Track the steps in synthetic or polynomial division.
Step-by-step division:
a. Leading term:
[tex]\[
\frac{x^3}{x} = x^2
\][/tex]
Multiply [tex]\( x^2 \)[/tex] by [tex]\( (x - 2) \)[/tex]:
[tex]\[
x^3 - 2x^2
\][/tex]
Subtract from the original polynomial:
[tex]\[
(x^3 - kx^2 + 2x - 4) - (x^3 - 2x^2) = (-k + 2)x^2 + 2x - 4
\][/tex]
b. Next term:
[tex]\[
\frac{(-k + 2)x^2}{x} = (-k + 2)x
\][/tex]
Multiply [tex]\((-k + 2)x\)[/tex] by [tex]\( (x - 2) \)[/tex]:
[tex]\[
(-k + 2)x^2 - 2(-k + 2)x = (-k + 2)x^2 + 2k - 4x
\][/tex]
Subtract:
[tex]\[
((-k + 2)x^2 + 2x - 4) - ((-k + 2)x^2 + 2k - 4x) = (2 - (-2k + 4))x - 4
= 2x - 4k + 4x - 4
= (2 + 4)x - 4k - 4
= (4)x - 4k - 4
\][/tex]
c. Last term:
[tex]\[
\frac{2x}{x} + 4 = 6 - k + 2
\frac{
The quotient is:
\(
Q(x) = x^2 + (-k + 2)x + 6
\)
4. Remainder:
We found that the remainder when dividing \( x^3 - kx^2 + 2x - 4 \) by \( x - 2 \) can be expressed as:
\[
R = 8 - 4k
\][/tex]
We want this remainder to be zero:
[tex]\[
8 - 4k = 0
\][/tex]
5. Solve for [tex]\( k \)[/tex]:
[tex]\[
8 = 4k \quad \Rightarrow \quad k = \frac{8}{4} = 2
\][/tex]
So, the value of [tex]\( k \)[/tex] that ensures the remainder is zero is:
[tex]\[
k = 2
\][/tex]
