Which option gives the linear factors of the polynomial 2x3−24x2+64x ?(1 point) Responses (x−8)(x−4) open paren x minus 8 close paren times open paren x minus 4 close paren 2x(x−8)(x−4) 2 x open paren x minus 8 close paren open paren x minus 4 close paren x(x−8)(x−4)

Answer 1

To find the linear factors of the polynomial \(2x^3 - 24x^2 + 64x\), we can first factor out the greatest common factor (GCF), which is \(2x\):

\[ 2x^3 - 24x^2 + 64x = 2x(x^2 - 12x + 32) \]

Next, we need to factor the quadratic expression \(x^2 - 12x + 32\). We look for two numbers that multiply to \(32\) and add to \(-12\). These numbers are \(-4\) and \(-8\). Therefore, we can factor the quadratic as follows:

\[ x^2 - 12x + 32 = (x - 4)(x - 8) \]

Putting everything together, we have:

\[ 2x^3 - 24x^2 + 64x = 2x(x - 4)(x - 8) \]

Thus, the complete factorization of the polynomial in terms of linear factors is:

\[ 2x(x - 4)(x - 8) \]

From the options given, the correct response is:

2x(x - 8)(x - 4)