To find the factored form of the expression 8x³ - 8x² - 30x, we can first factor out the greatest common factor of 2x to get:
8x³ - 8x² - 30x = 2x(4x² - 4x - 15)
Next, we can factor the quadratic expression inside the parentheses using either factoring by grouping, completing the square, or the quadratic formula. In this case, we can use factoring by grouping:
2x(4x² - 4x - 15) = 2x(2x + 3)(2x - 5)
Therefore, the factored form of the expression is:
8x³ - 8x² - 30x = 2x(2x + 3)(2x - 5)
Explanation:
We are given the expression 8x³ - 8x² - 30x and we want to factor it into its simplest form. To do this, we can follow these steps:
Step 1: Factor out the Greatest Common Factor (GCF)
The first step is to factor out the greatest common factor (GCF) of the terms in the expression. In this case, the GCF is 2x, so we can factor out 2x from each term:
8x³ - 8x² - 30x = 2x(4x² - 4x - 15)
Notice that when we factor out 2x from each term, we are left with the expression 4x² - 4x - 15 inside the parentheses.
Step 2: Factor the quadratic expression inside the parentheses
The next step is to factor the quadratic expression 4x² - 4x - 15 inside the parentheses. There are different methods for factoring quadratics, such as factoring by grouping, completing the square, or using the quadratic formula. In this case, we can use factoring by grouping.
To factor 4x² - 4x - 15 by grouping, we need to find two numbers that multiply to -15 and add up to -4x. These numbers are -5 and 3, since (-5) * 3 = -15 and (-5) + 3 = -2. So we can rewrite the expression as:
4x² - 4x - 15 = 4x² - 5x + x - 15
= x(4x - 5) + 3(4x - 5)
= (4x - 5)(x + 3)
So we have factored 4x² - 4x - 15 into (4x - 5)(x + 3).
Step 3: Write the factored form of the expression
Finally, we can substitute the factored form of 4x² - 4x - 15 back into the expression we factored in step 1:
8x³ - 8x² - 30x = 2x(4x² - 4x - 15)
= 2x(4x - 5)(x + 3)
So the factored form of 8x³ - 8x² - 30x is 2x(4x - 5)(x + 3).
i hope this was helpful