factor by grouping
5x^3-x^2-45x-9=0
5x^3-x^2-45x-9=0
x^2(5x - 1) -9(5x + 1) = 0
I think you have a typo, and my guess is that the -9 is +9
that would make it
x^2(5x - 1) -9(5x - 1) = 0
(5x-1)(x^2 - 9) = 0
(5x-1)(x+3)(x-3) = 0
x = 1/5, -3, +3
To factor by grouping, you can rearrange the terms in a way that allows you to group them together in pairs. Let's factor the given equation 5x^3 - x^2 - 45x - 9 = 0 using this method.
Step 1: Group the terms
Start by grouping the first two terms and the last two terms:
(5x^3 - x^2) + (-45x - 9) = 0
Step 2: Factor out the greatest common factor
Next, factor out the greatest common factor separately from each group. In the first group, the greatest common factor is x^2, and in the second group, the greatest common factor is 9:
x^2(5x - 1) + 9(-5x - 1) = 0
Step 3: Simplify the expression
Distribute the factors on both groups:
5x^3 - x^2 - 45x - 9 = 0
Step 4: Group the factored terms
Now, notice that we have common factors in each group, (5x - 1) in the first group and (-5x - 1) in the second group. We can simplify this as follows:
x^2(5x - 1) - 9(5x - 1) = 0
Step 5: Factor out the common binomial factor
Now we have a common binomial factor, (5x - 1), in both terms:
(5x - 1)(x^2 - 9) = 0
Step 6: Further factorization
Finally, the expression can be factored completely:
(5x - 1)(x + 3)(x - 3) = 0
Therefore, the factored form of the equation 5x^3 - x^2 - 45x - 9 = 0 is (5x - 1)(x + 3)(x - 3) = 0.