Factor the polynomial completely.
6x^3 + 36x^2 - 240x
6x(x^2+6x-30)
6x(x+3-sqrt39)(x+3+sqrt39)
To factor the polynomial 6x^3 + 36x^2 - 240x completely, follow these steps:
Step 1: Look for the greatest common factor (GCF) among the terms.
In this case, the GCF is 6x since it can be factored out of each term:
6x^3 + 36x^2 - 240x = 6x(x^2 + 6x - 40)
Step 2: Next, focus on factoring the quadratic trinomial x^2 + 6x - 40.
To factor this quadratic trinomial, you need to find two binomials whose product is equal to the trinomial.
Step 3: Find the factor pairs of the constant term (-40) that add up to the coefficient of the linear term (6).
The factor pairs of -40 are (-1,40), (1,-40), (-2,20), (2,-20), (-4,10), (4,-10), (-5,8), and (5,-8).
Among these pairs, the pair (-5,8) adds up to 3, which is closest to the coefficient of the linear term (6).
Step 4: Rewrite the trinomial using the factors found in the previous step:
x^2 + 6x - 40 = (x - 5)(x + 8)
Step 5: Bring the factor from Step 1 back into the factored trinomial:
6x(x^2 + 6x - 40) = 6x(x - 5)(x + 8)
Therefore, the completely factored form of the polynomial 6x^3 + 36x^2 - 240x is:
6x(x - 5)(x + 8)