Which option gives the linear factors of the polynomial 2x3−24x2+64x

Question

2
𝑥
3
−
24
𝑥
2
+
64
𝑥
?(1 point)
Responses

(x−8)(x−4)
(
𝑥
−
8
)
(
𝑥
−
4
)
open paren x minus 8 close paren times open paren x minus 4 close paren

x(x−8)(x−4)
𝑥
(
𝑥
−
8
)
(
𝑥
−
4
)
x open paren x minus 8 close paren open paren x minus 4 close paren

2x(x−8)(x−4)
2
𝑥
(
𝑥
−
8
)
(
𝑥
−
4
)
2 x open paren x minus 8 close paren open paren x minus 4 close paren

Answer 1

To find the linear factors of the polynomial \(2x^3 - 24x^2 + 64x\), let's first factor it.

1. Factor out the greatest common factor (GCF): The GCF of the terms is \(2x\): \[ 2x(x^2 - 12x + 32) \]

2. Now factor the quadratic \(x^2 - 12x + 32\): We need to find two numbers that multiply to \(32\) and add to \(-12\). These numbers are \(-8\) and \(-4\): \[ x^2 - 12x + 32 = (x - 8)(x - 4) \]

3. Putting it all together: Thus, the complete factorization of the polynomial is: \[ 2x(x - 8)(x - 4) \]

From the options provided, the correct factorization of the polynomial \(2x^3 - 24x^2 + 64x\) in linear factors is: \[ \boxed{2x(x−8)(x−4)} \]