To determine the value of the constant [tex]\( k \)[/tex] in the factored form of the polynomial [tex]\( 8x^3 + 6x^2 - 32x - 24 \)[/tex], we need to expand the factored form and compare the coefficients.
We are given the polynomial:
[tex]\[ 8x^3 + 6x^2 - 32x - 24 \][/tex]
Its factored form is given as:
[tex]\[ 2(x+k)(x-k)(4x+3) \][/tex]
Let's expand this factored form step by step:
1. Begin by expanding [tex]\( (x+k)(x-k) \)[/tex]:
[tex]\[ (x+k)(x-k) = x^2 - k^2 \][/tex]
2. Multiply this result by [tex]\( 4x+3 \)[/tex]:
[tex]\[ (x^2 - k^2)(4x + 3) = x^2(4x + 3) - k^2(4x + 3) \][/tex]
[tex]\[ = 4x^3 + 3x^2 - 4k^2x - 3k^2 \][/tex]
3. Finally, multiply by 2:
[tex]\[ 2(4x^3 + 3x^2 - 4k^2x - 3k^2) = 8x^3 + 6x^2 - 8k^2x - 6k^2 \][/tex]
Now, compare the expanded form with the given polynomial:
[tex]\[ 8x^3 + 6x^2 - 8k^2x - 6k^2 \quad \text{matches} \quad 8x^3 + 6x^2 - 32x - 24 \][/tex]
By matching the coefficients, we get:
1. For the [tex]\( x \)[/tex] term:
[tex]\[ -8k^2 = -32 \][/tex]
[tex]\[ k^2 = 4 \][/tex]
[tex]\[ k = \pm 2 \][/tex]
Since [tex]\( k \)[/tex] can be either 2 or -2, but we are often interested in the absolute value of [tex]\( k \)[/tex], which is:
[tex]\[ k = 2 \][/tex]
Therefore, the correct value of the constant [tex]\( k \)[/tex] is:
[tex]\[ \boxed{2} \][/tex]
