Question

Solve the polynomial equation by using the root​ method, and check the solutions graphically.
3 x cubed minus 24 equals 0
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Part 1
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​(Simplify your answer. Use a comma to separate answers as​ needed.)

Answer 1

To solve the polynomial equation \( 3x^3 - 24 = 0 \) using the root method, we can follow these steps:

1. **Isolate the \( x^3 \) term:**
\[
3x^3 = 24
\]

2. **Divide both sides by 3:**
\[
x^3 = 8
\]

3. **Take the cube root of both sides:**
\[
x = \sqrt[3]{8}
\]
Since \( \sqrt[3]{8} = 2 \), we get:
\[
x = 2
\]

The polynomial \( 3x^3 - 24 \) is a cubic equation, and it has one real root and two complex roots since a cubic equation always has three roots (counting multiplicities). The real root we've found is \( x = 2 \).

To find the other roots, we can factor the polynomial:
\[
3(x^3 - 8) = 3(x - 2)(x^2 + 2x + 4) = 0
\]
The quadratic \( x^2 + 2x + 4 \) can be solved using the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
For our quadratic \( a = 1, b = 2, c = 4 \):
\[
x = \frac{-2 \pm \sqrt{2^2 - 4 \cdot 1 \cdot 4}}{2 \cdot 1} = \frac{-2 \pm \sqrt{4 - 16}}{2} = \frac{-2 \pm \sqrt{-12}}{2} = \frac{-2 \pm 2i\sqrt{3}}{2} = -1 \pm i\sqrt{3}
\]

### Summary of Roots:
1. The real root: \( x = 2 \)
2. Two complex roots: \( x = -1 + i\sqrt{3} \) and \( x = -1 - i\sqrt{3} \)

**Final answer:**
The solution to the polynomial \( 3x^3 - 24 = 0 \) is:
\[
x = 2, -1 + i\sqrt{3}, -1 - i\sqrt{3}
\]

### Graphically Checking the Solutions:
You can plot the function \( f(x) = 3x^3 - 24 \) using graphing tools or calculators. The graph will intersect the x-axis at \( x = 2 \), confirming that it is the real root. The other two roots will not be on the graph as they are complex numbers.