In the polynomial function below, [tex]\( k \)[/tex] is a constant:

[tex]\[ f(x) = 3x^3 - 8x^2 + 5x - k \][/tex]

If [tex]\( (x - 2) \)[/tex] is a factor of [tex]\( f(x) \)[/tex], what is the value of [tex]\( k \)[/tex]?



Answer :

To determine the value of [tex]\( k \)[/tex] such that [tex]\((x - 2)\)[/tex] is a factor of the polynomial [tex]\( f(x) = 3x^3 - 8x^2 + 5x - k \)[/tex], we can use the Factor Theorem. The Factor Theorem states that if [tex]\( (x - a) \)[/tex] is a factor of a polynomial [tex]\( f(x) \)[/tex], then [tex]\( f(a) = 0 \)[/tex].

Given that [tex]\((x-2)\)[/tex] is a factor, it means that substituting [tex]\( x = 2 \)[/tex] into the polynomial should result in the polynomial being equal to 0. Therefore, we need to find [tex]\( k \)[/tex] such that:

[tex]\[ f(2) = 0 \][/tex]

Substitute [tex]\( x = 2 \)[/tex] into the polynomial [tex]\( f(x) \)[/tex]:

[tex]\[ f(2) = 3(2)^3 - 8(2)^2 + 5(2) - k \][/tex]

Calculate each term separately:

[tex]\[ 2^3 = 8 \implies 3(8) = 24 \][/tex]
[tex]\[ 2^2 = 4 \implies 8(4) = 32 \][/tex]
[tex]\[ 5(2) = 10 \][/tex]

Now, substitute these values back into the equation:

[tex]\[ f(2) = 24 - 32 + 10 - k \][/tex]

Combine the constants:

[tex]\[ 24 - 32 + 10 = 2 \][/tex]

Therefore, we have:

[tex]\[ f(2) = 2 - k \][/tex]

Since [tex]\((x-2)\)[/tex] is a factor, [tex]\( f(2) = 0 \)[/tex]. Hence, we set up the equation:

[tex]\[ 2 - k = 0 \][/tex]

Solve for [tex]\( k \)[/tex]:

[tex]\[ k = 2 \][/tex]

Thus, the value of [tex]\( k \)[/tex] that makes [tex]\((x-2)\)[/tex] a factor of [tex]\( f(x) = 3x^3 - 8x^2 + 5x - k \)[/tex] is:

[tex]\[ \boxed{2} \][/tex]