Which statement describes the graph of [tex]f(x)=-x^4+3 x^3+10 x^2[/tex]?

A. The graph crosses the [tex]x[/tex]-axis at [tex]x=0[/tex] and touches the [tex]x[/tex]-axis at [tex]x=5[/tex] and [tex]x=-2[/tex].
B. The graph touches the [tex]x[/tex]-axis at [tex]x=0[/tex] and crosses the [tex]x[/tex]-axis at [tex]x=5[/tex] and [tex]x=-2[/tex].
C. The graph crosses the [tex]x[/tex]-axis at [tex]x=0[/tex] and touches the [tex]x[/tex]-axis at [tex]x=-5[/tex] and [tex]x=2[/tex].
D. The graph touches the [tex]x[/tex]-axis at [tex]x=0[/tex] and crosses the [tex]x[/tex]-axis at [tex]x=-5[/tex] and [tex]x=2[/tex].

The graph of the function f ( x ) = − x 4 + 3 x 3 + 10 x 2 touches the x-axis at x = 0 and crosses the x-axis at x = 5 and x = − 2 . Therefore, the correct answer is A. The graph crosses the x-axis at x=0 and touches the x-axis at x=5 and x=-2.
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Answered by Anonymous | 2025-07-04

Factor the polynomial f ( x ) = − x 4 + 3 x 3 + 10 x 2 to find its roots: f ( x ) = − x 2 ( x − 5 ) ( x + 2 ) .
Identify the roots and their multiplicities: x = 0 (multiplicity 2), x = 5 (multiplicity 1), and x = − 2 (multiplicity 1).
Determine the behavior at each root based on its multiplicity: touches at x = 0 , crosses at x = 5 and x = − 2 .
The graph touches the x -axis at x = 0 and crosses the x -axis at x = 5 and x = − 2 . The graph touches the x -axis at x = 0 and crosses the x -axis at x = 5 and x = − 2. ​

Explanation

Problem Analysis We are given the function f ( x ) = − x 4 + 3 x 3 + 10 x 2 and asked to determine the behavior of its graph at its x-intercepts. Specifically, we need to find out whether the graph crosses or touches the x-axis at each x-intercept.

Finding the x-intercepts To determine the x-intercepts, we need to find the roots of the function f ( x ) . We can do this by setting f ( x ) = 0 and solving for x . So, we have − x 4 + 3 x 3 + 10 x 2 = 0 .

Factoring the function We can factor out an x 2 from the equation: x 2 ( − x 2 + 3 x + 10 ) = 0 . Now, we need to factor the quadratic − x 2 + 3 x + 10 . We can rewrite it as − ( x 2 − 3 x − 10 ) . Factoring the quadratic x 2 − 3 x − 10 , we look for two numbers that multiply to -10 and add to -3. These numbers are -5 and 2. So, x 2 − 3 x − 10 = ( x − 5 ) ( x + 2 ) . Therefore, the factored form of the function is f ( x ) = − x 2 ( x − 5 ) ( x + 2 ) .

Identifying the roots and their multiplicities Now we can identify the roots of the function. The roots are x = 0 , x = 5 , and x = − 2 . The root x = 0 comes from the factor x 2 , which has a multiplicity of 2. The roots x = 5 and x = − 2 come from the factors ( x − 5 ) and ( x + 2 ) , respectively, each with a multiplicity of 1.

Determining the behavior at each root The behavior of the graph at each root depends on the multiplicity of the root. If the multiplicity is even, the graph touches the x-axis at that root. If the multiplicity is odd, the graph crosses the x-axis at that root. Since the root x = 0 has a multiplicity of 2 (even), the graph touches the x-axis at x = 0 . The roots x = 5 and x = − 2 have multiplicities of 1 (odd), so the graph crosses the x-axis at x = 5 and x = − 2 .

Conclusion Therefore, the graph touches the x-axis at x = 0 and crosses the x-axis at x = 5 and x = − 2 .


Examples
Understanding the behavior of polynomial functions, such as where they cross or touch the x-axis, is crucial in various real-world applications. For instance, in engineering, when designing a bridge, engineers use polynomial functions to model the load distribution. The points where the function crosses the x-axis (roots) can represent critical stress points. Similarly, in economics, polynomial functions can model cost and revenue curves, where the roots indicate break-even points. Analyzing these functions helps engineers and economists make informed decisions about safety and profitability.

Answered by GinnyAnswer | 2025-07-04