Question

What is the completely factored form of this polynomial? 2x5 + 12x3 − 54x

Answer 1

Answer:

$2x(x^2-3)(x^2+9)$

Step-by-step explanation:

Given polynomial:

$2x^5+12x^3-54x$

Factor out the common term $2x$:

$\implies 2x(x^4+6x^2-27)$

To factor the trinomial $x^4+6x^2-27$:

$\textsf{Let }u=x^2 \implies u^2+6u-27$

Factor the quadratic by finding two numbers that multiply to -27 and sum to 6:  9 and -3

Rewrite the middle term as the sum of these two numbers:

$\implies u^2+9u-3u-27$

Factorize the first two terms and the last two terms separately:

$\implies u(u+9)-3(u+9)$

Factor out the common term (u + 9):

$\implies (u-3)(u+9)$

Substitute back $u=x^2$:

$\implies (x^2-3)(x^2+9)$

Therefore, the factored form of the given polynomial is:

$\implies 2x(x^2-3)(x^2+9)$

Answer 2

Answer:

2x(x^2 - 3)(x^2 + 9)

Step-by-step explanation:

2x^5 + 12x^3 − 54x

2x(x^4 + 6x - 27)

Since -3 + 9 = 6 and -3 x 9 = -27:

2x(x^2 - 3)(x^2 + 9)