select the correct answer.\nwhat is the completely factored form of this polynomial?\n2x^5 + 12x^3…

select the correct answer.\nwhat is the completely factored form of this polynomial?\n2x^5 + 12x^3 - 54x\n\na. 2x(x^2 + 3)(x + 9)(x - 9)\nb. 2x(x - 3)(x + 9)\nc. 2x(x^2 + 3)(x + 3)(x - 3)\nd. 2x(x^2 - 3)(x^2 + 9)

select the correct answer.\nwhat is the completely factored form of this polynomial?\n2x^5 + 12x^3 - 54x\n\na. 2x(x^2 + 3)(x + 9)(x - 9)\nb. 2x(x - 3)(x + 9)\nc. 2x(x^2 + 3)(x + 3)(x - 3)\nd. 2x(x^2 - 3)(x^2 + 9)

Answer

Explanation:

Step1: Factor out the GCF

First, find the greatest - common factor of the terms (2x^{5}), (12x^{3}), and (-54x). The GCF of (2), (12), and (- 54) is (2), and the GCF of (x^{5}), (x^{3}), and (x) is (x). So, (2x^{5}+12x^{3}-54x = 2x(x^{4}+6x^{2}-27)).

Step2: Let (u = x^{2})

Substitute (u) into the polynomial (x^{4}+6x^{2}-27), we get (u^{2}+6u - 27).

Step3: Factor the quadratic in (u)

Factor (u^{2}+6u - 27). We need to find two numbers that multiply to (-27) and add up to (6). The numbers are (9) and (-3). So, (u^{2}+6u - 27=(u + 9)(u - 3)).

Step4: Substitute back (u=x^{2})

Replace (u) with (x^{2}), we have (x^{4}+6x^{2}-27=(x^{2}+9)(x^{2}-3)).

Step5: Write the completely - factored form

The completely - factored form of (2x^{5}+12x^{3}-54x) is (2x(x^{2}+9)(x^{2}-3)).

Answer:

D. (2x(x^{2}-3)(x^{2}+9))