What are the linear and quadratic factors of 8x4−32x3−5x2+20x

Question

What are the linear and quadratic factors of  8x4−32x3−5x2+20x
8
𝑥
4
−
32
𝑥
3
−
5
𝑥
2
+
20
𝑥
 ?(1 point)
Responses

x(x−4)(8x2+5)
𝑥
(
𝑥
−
4
)
(
8
𝑥
2
+
5
)
x open paren x minus 4 close paren open paren 8 x squared plus 5 close paren

x(x−4)(8x2−5)
𝑥
(
𝑥
−
4
)
(
8
𝑥
2
−
5
)
x open paren x minus 4 close paren open paren 8 x squared minus 5 close paren

x(x+4)(8x2−5)

GPT-4o mini · Answer

To factor the polynomial $ 8x^4 - 32x^3 - 5x^2 + 20x $, we start by looking for common factors. 1. **Factor out the greatest common factor (GCF):** The GCF of the terms is $ x $. So, we can factor $ x $ out: $$ 8x^4 - 32x^3 - 5x^2 + 20x = x(8x^3 - 32x^2 - 5x + 20) $$ 2. **Factor the cubic polynomial $ 8x^3 - 32x^2 - 5x + 20 $:** We can use the rational root theorem or synthetic division to find potential rational roots. Testing $ x = 4 $: $$ 8(4)^3 - 32(4)^2 - 5(4) + 20 = 512 - 512 - 20 + 20 = 0 $$ Since $ x = 4 $ is a root, we can factor $ 8x^3 - 32x^2 - 5x + 20 $ using synthetic division by $ x - 4 $. Performing synthetic division, we divide $ 8x^3 - 32x^2 - 5x + 20 $ by $ x - 4 $: ``` 4 | 8 -32 -5 20 | 32 0 -20 ----------------- 8 0 -5 0 ``` The result is: $$ 8x^2 - 5 $$ So, $$ 8x^3 - 32x^2 - 5x + 20 = (x - 4)(8x^2 - 5) $$ 3. **Combining everything together:** $$ x(8x^3 - 32x^2 - 5x + 20) = x(x - 4)(8x^2 - 5) $$ Thus, the complete factorization of the original polynomial $ 8x^4 - 32x^3 - 5x^2 + 20x $ is: $$ x(x - 4)(8x^2 - 5) $$ **Final answer:** $ x(x−4)(8x^2−5) $ is the correct choice.