help fact the trinomial completely and show how it was done.
12x^3 - 52x^2 -40x
4x(3x^2-13x-10)
4x(3x+2)(x-5)
To factor the trinomial 12x^3 - 52x^2 - 40x completely, you can follow these steps:
Step 1: Find the greatest common factor (GCF) among the terms, if any.
In this case, there isn't a common factor that can be factored out from all the terms.
Step 2: Look for common factors among pairs of terms.
One common factor we can find is 4x, which can be factored from all three terms:
12x^3 - 52x^2 - 40x = 4x(3x^2 - 13x - 10)
Step 3: Factor the remaining quadratic trinomial.
To factor the quadratic trinomial 3x^2 - 13x - 10, we are looking for two binomials with the following form:
(ax + b)(cx + d)
The coefficient of x^2 is 3, so ac = 3. The constant term is -10, so bd = -10.
We need to find factors of -10 that sum up to -13 (coefficient of x).
The factors are -10 and 1, so we can rewrite the trinomial as:
3x^2 - 10x + x - 10
Now, we group the terms with common factors:
x(3x - 10) + 1(3x - 10)
We notice that both terms have a common factor of (3x - 10):
(3x - 10)(x + 1)
Step 4: Put all the factors together.
Now, we can combine the factor found in step 2 with the factor found in step 3:
4x(3x - 10)(x + 1)
Therefore, the trinomial 12x^3 - 52x^2 - 40x completely factors to:
4x(3x - 10)(x + 1)