Solve an equation 8x^3+4x^2-18x-9=0 algebraically for all values of x.

try grouping

8x^3+4x^2-18x-9=0
4x^2(2x+1) - 9(2x+1) = 0
(2x+1)(4x^2-9) = 0
(2x-1)(2x-3)(2x+3) = 0
x = 1/2, 3/2, -3/2

Thank you so much!

please help this is due Monday!

To solve the equation 8x^3 + 4x^2 - 18x - 9 = 0 algebraically for all values of x, we can use various algebraic techniques. One technique is to factor the equation if possible.

1. Begin by examining if the equation has any common factors. In this case, there are no common factors among the coefficients.

2. Since factoring doesn't seem straightforward, we can try to find rational roots using the Rational Root Theorem. According to the theorem, any rational root of the equation should have the following form: ± a factor of the constant term (9) divided by a factor of the leading coefficient (8).

In this case, the factors of 9 are ±1, ±3, and ±9. The factors of 8 are ±1, ±2, ±4, and ±8. So, the possible rational roots are ±1/1, ±3/1, ±9/1, ±1/2, ±3/2, ±9/2, ±1/4, ±3/4, ±9/4, ±1/8, ±3/8, ±9/8.

3. Now, we need to test each of these possible rational roots using synthetic division or long division to see if they are the actual solutions of the equation.

Let's take the first possible root, x = -1:

Using synthetic division:
```
-1 | 8 4 -18 -9
| -8 4 14
---------------
8 -4 -14 5
```

The remainder is 5, so -1 is not a root. We need to repeat this process for the other possible roots.

After testing all the possible rational roots, we find that there are no rational roots for this equation. Hence, it cannot be factored easily using rational numbers.

4. At this point, it is helpful to use numerical methods or calculators to approximate the roots of the equation. One common numerical method is Newton's method or using a graphing calculator.

Using a graphing calculator or a software tool, we find that the roots of the equation are approximately x = -1.326, x ≈ 0.738, and x ≈ 1.589.

Therefore, the algebraic solution for the equation 8x^3 + 4x^2 - 18x - 9 = 0 for all values of x is x ≈ -1.326, x ≈ 0.738, and x ≈ 1.589.