Given the polynomial 8x^3+6x^2-32x-24 what is the value of the constant 'K'in the factored form? 8x^3+6x^2-32x-24=2(x+k)(x-k)(4x+3) 4 3 2 1

To find the value of the constant 'K' in the factored form of the polynomial $8x^{3}+6x^{2}-32x-24$, we need to expand the factored form and compare it with the original polynomial.<br /><br />The factored form is given as:<br />$8x^{3}+6x^{2}-32x-24=2(x+k)(x-k)(4x+3)$<br /><br />Expanding the factored form:<br />$2(x+k)(x-k)(4x+3) = 2(x^2 - k^2)(4x+3) = 2(4x^3 + 3x^2 - 4kx^2 - 3k^2x) = 8x^3 + 6x^2 - 8kx^2 - 6k^2x = 8x^3 + 6x^2 - 32x - 24$<br /><br />Comparing the expanded form with the original polynomial, we can see that the constant term in the expanded form is $-6k^2$, which should be equal to the constant term in the original polynomial, which is $-24$.<br /><br />Therefore, we have:<br />$-6k^2 = -24$<br />$k^2 = 4$<br />$k = \pm 2$<br /><br />So, the value of the constant 'K' in the factored form is either $2$ or $-2$.