Given the functions f(x)=x^3 +5 and g(x)=5x^2 +x, solve the equation f(x)=g(x) and interpret the solutions. Which of the following statements is correct?

Question

1)The solution f(x)=g(x) is x=-1, x=1, and x=5. This means that f(x) and g(x) have three points of intersection.
2)The solution to f(x)=g(x) is x=-5/4 and x=1. This means that f(x) and g(x) each have two x-intercepts.
3) The solution to f(x)=g(x) is x=-1, x=1, and x=5. This means that f(x) and g(x) each have three x-intercepts.
4)The solutions to f(x)=g(x) is x=-5/4 and x=1. This means that f(x) and g(x) have two points of intersection.

Answer 1

To solve the equation \( f(x) = g(x) \), we start by setting the two functions equal to each other:

\[ x^3 + 5 = 5x^2 + x \]

We rearrange this equation to bring all terms to one side:

\[ x^3 - 5x^2 - x + 5 = 0 \]

Next, we will attempt to find the roots of this cubic equation. We can do this either by factoring (if applicable) or using numerical methods or graphing techniques. However, we can first check for rational roots using the Rational Root Theorem, or in this case, testing some potential roots:

Testing \( x = 1 \):

\[ 1^3 - 5(1^2) - 1 + 5 = 1 - 5 - 1 + 5 = 0 \]

Thus, \( x = 1 \) is a root. We can then divide the cubic polynomial by \( x - 1 \). Using synthetic division, we get:

\[ \begin{array}{r|rrrr} 1 & 1 & -5 & -1 & 5 \ & & 1 & -4 & -5 \ \hline & 1 & -4 & -5 & 0 \ \end{array} \]

This gives \( x^2 - 4x - 5 \), which we can factor:

\[ x^2 - 4x - 5 = (x - 5)(x + 1) \]

Thus, the original cubic can be written as:

\[ (x - 1)(x - 5)(x + 1) = 0 \]

This means the solutions are:

\[ x = 1, \quad x = 5, \quad x = -1 \]

Now let's interpret the solutions. These values mean that the graphs of \( f(x) \) and \( g(x) \) intersect at three points: \( (-1, f(-1)) \), \( (1, f(1)) \), and \( (5, f(5)) \).

Now let's analyze the provided statements:

1. The solution \( f(x) = g(x) \) is \( x = -1, 1, 5 \). This means that \( f(x) \) and \( g(x) \) have three points of intersection.

   • This statement is true.

2. The solution to \( f(x) = g(x) \) is \( x = -5/4 \) and \( x = 1 \). This means that \( f(x) \) and \( g(x) \) each have two x-intercepts.

   • This statement is incorrect.

3. The solution to \( f(x) = g(x) \) is \( x = -1, 1, 5 \). This means that \( f(x) \) and \( g(x) \) each have three x-intercepts.

   • This statement is misleading because while it states the correct solutions, it implies that \( f(x) \) and \( g(x) \) each have three x-intercepts, which is not necessarily true.

4. The solutions to \( f(x) = g(x) \) are \( x = -5/4 \) and \( x = 1 \). This means that \( f(x) \) and \( g(x) \) have two points of intersection.

   • This statement is incorrect.

Thus, the correct answer is:

Option 1: The solution \( f(x) = g(x) \) is \( x = -1, x = 1, x = 5 \). This means that \( f(x) \) and \( g(x) \) have three points of intersection.