Arlana graphed the system of equations that can be used to solve x cubed minus 5 x squared + 2 = negative x cubed + 17 x.

On a coordinate plane, 2 functions are shown. They intersect at (negative 2, negative 26), (0.11, 1.94), and (4.39, negative 9.81).

What are the roots of the polynomial equation? Round noninteger roots to the nearest hundredth.
–26, –9.81, 1.94
–26
–2
–2, 0.11, 4.39

To find the roots of the polynomial equation, we need to identify the x-values where the two given functions intersect. These intersections correspond to the roots of the polynomial.

Given the system of equations:
\[ f(x) = x^3 - 5x^2 + 2 \]
\[ g(x) = -x^3 + 17x \]

The intersections of these functions are at the points:
\[ (-2, -26), \, (0.11, 1.94), \, (4.39, -9.81) \]

The x-values of these intersection points are \(-2\), \(0.11\), and \(4.39\).

These x-values are the roots of the polynomial equation derived from setting \(f(x) = g(x)\):
\[ x^3 - 5x^2 + 2 = -x^3 + 17x \]

Rewriting it:
\[ x^3 - 5x^2 + 2 + x^3 - 17x = 0 \]
\[ 2x^3 - 5x^2 - 17x + 2 = 0 \]

Therefore, the roots of the polynomial equation \(2x^3 - 5x^2 - 17x + 2 = 0\) are:
\[ -2, \, 0.11, \, 4.39 \]

These match the x-coordinates of the intersection points.

So the correct answer is:
\[ -2, 0.11, 4.39 \]

Do you want more details or have any questions?

Here are five related questions for further exploration:
1. How do you derive a polynomial equation from a system of equations?
2. What are the steps to solve a cubic equation?
3. How can you verify the roots of a polynomial equation?
4. What methods can be used to find intersection points of two functions graphically?
5. How does the behavior of polynomial functions change with higher degrees?

**Tip:** When solving polynomial equations, always check for possible factoring methods or use the Rational Root Theorem to identify potential rational roots.

Arlana graphed the system of equations that can be used to solve x cubed minus 5 x squared + 2 = negative x cubed + 17 x. On a coordinate plane, 2 functions are shown. They intersect at (negative 2, negative 26), (0.11, 1.94), and (4.39, negative 9.81). What are the roots of the polynomial equation? Round noninteger roots to the nearest hundredth. –26, –9.81, 1.94 –26 –2 –2, 0.11, 4.39

Learn how to find the roots of the cubic equation derived from the system x^3 - 5x^2 + 2 = -x^3 + 17x by identifying the intersection points of the two functions. The problem includes key concepts in algebra and graphing polynomial equations. Suitable for students in Grades 10-12.

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