Solve the polynomial equation by using the root​ method, and check the solutions graphically.

Question

3 x cubed minus 24 equals 0
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Part 1
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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.