Answered

The following system of equations can be used to find the roots of the equation $x^3+72=5 x^2+18 x$.

$\begin{array}{l}
y=x^3+72 \\
y=5 x^2+18 x
\end{array}$



Graph this system of equations on the graphing calculator. Adjust the window so you can find all of the points of intersection for the system of equations.



What are the roots of the original polynomial equation? Choose three correct answers.

  • $8$
  • $0$

Verified Answer :

  • Equate the system \(y = x^3 + 72\) and \(y = 5x^2 + 18x\) to form the cubic equation \(x^3 - 5x^2 - 18x + 72 = 0\).
  • Use the Rational Root Theorem to identify the first root, \(x = -4\).
  • Divide the cubic polynomial by \((x + 4)\) to obtain the quadratic \(x^2 - 9x + 18\).
  • Factor the quadratic as \((x - 3)(x - 6)\) to find the remaining roots, \(3\) and \(6\).
  • The three roots are \(\boxed{-4, 3, 6}\).

Explanation

  1. Analyzing the System of Equations

To find the roots of the given system, we start by equating the two expressions for \(y\):
\[y = x^3 + 72\]
\[y = 5x^2 + 18x\]
Setting them equal to each other gives us the polynomial equation:
\[x^3 + 72 = 5x^2 + 18x\]
Rearranging all terms to one side, we obtain the cubic equation:
\[x^3 - 5x^2 - 18x + 72 = 0\]

  1. Finding the Roots of the Polynomial

We seek values of \(x\) that satisfy the equation \(x^3 - 5x^2 - 18x + 72 = 0\). Using the Rational Root Theorem, we test potential factors of the constant term 72 (such as \(\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \dots\)). Testing \(x = -4\):
\[(-4)^3 - 5(-4)^2 - 18(-4) + 72 = -64 - 5(16) + 72 + 72 = -64 - 80 + 144 = -144 + 144 = 0\]
Since \(x = -4\) is a root, \((x + 4)\) must be a factor of the polynomial. Dividing \(x^3 - 5x^2 - 18x + 72\) by \((x + 4)\) using synthetic division or long division yields the quadratic factor:
\[x^2 - 9x + 18\]
We can now factor the resulting quadratic expression:
\[x^2 - 9x + 18 = (x - 3)(x - 6)\]
Therefore, the polynomial can be written as \((x + 4)(x - 3)(x - 6) = 0\).

  1. Conclusion and Final Roots

The roots of the equation \(x^3 - 5x^2 - 18x + 72 = 0\) are the values of \(x\) that make each linear factor equal to zero:
\(x + 4 = 0 \implies x = -4\)
\(x - 3 = 0 \implies x = 3\)
\(x - 6 = 0 \implies x = 6\)
Thus, the three roots are \(-4, 3, \text{ and } 6\). These values represent the x-coordinates where the graphs of \(y = x^3 + 72\) and \(y = 5x^2 + 18x\) intersect. The final roots are \(\boxed{-4, 3, 6}\).

Examples

In physics, when modeling the motion of two objects, you might find their positions at time \(t\) described by different functions. Determining when they are at the same position is equivalent to finding the intersection points of their trajectory equations, which often results in solving a system of polynomial equations. By identifying the roots of the resulting equation, you can pinpoint the specific times or locations where the objects meet, allowing for precise collision prediction or synchronization analysis.

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