Certainly! To find the value of [tex]\( k \)[/tex] such that [tex]\( x-3 \)[/tex] is a factor of the polynomial [tex]\( x^3 - kx^2 + x - 6 \)[/tex], we can use the Factor Theorem. The Factor Theorem states that [tex]\( x-3 \)[/tex] is a factor of a polynomial if and only if substituting [tex]\( x = 3 \)[/tex] into the polynomial gives zero.
Let's substitute [tex]\( x = 3 \)[/tex] into the polynomial and set it equal to zero:
[tex]\[ P(x) = x^3 - kx^2 + x - 6 \][/tex]
Substituting [tex]\( x = 3 \)[/tex]:
[tex]\[ P(3) = 3^3 - k(3^2) + 3 - 6 \][/tex]
Now, simplify it step-by-step:
1. Calculate [tex]\( 3^3 \)[/tex]:
[tex]\[ 3^3 = 27 \][/tex]
2. Calculate [tex]\( k \cdot 3^2 \)[/tex]:
[tex]\[ k \cdot 3^2 = 9k \][/tex]
3. Calculate the remaining constants ( [tex]\( 3 \)[/tex] and [tex]\( -6 \)[/tex] ):
[tex]\[ 3 - 6 = -3 \][/tex]
Now, combine these steps:
[tex]\[ 27 - 9k + 3 - 6 = 0 \][/tex]
Simplify the equation:
[tex]\[ 27 - 9k - 3 = 0 \][/tex]
[tex]\[ 24 - 9k = 0 \][/tex]
Solving for [tex]\( k \)[/tex]:
[tex]\[ 24 = 9k \][/tex]
[tex]\[ k = \frac{24}{9} \][/tex]
[tex]\[ k = \frac{8}{3} \][/tex]
Therefore, the value of [tex]\( k \)[/tex] that makes [tex]\( x-3 \)[/tex] a factor of [tex]\( x^3 - kx^2 + x - 6 \)[/tex] is:
[tex]\[ k = \frac{8}{3} \][/tex]
