To solve the equation \( f(x) = g(x) \), we need to set the two functions equal to each other:
\[ f(x) = g(x) \implies x^3 + 5 = 5x^2 + x \]
Rearranging this equation gives:
\[ x^3 - 5x^2 - x + 5 = 0 \]
We can attempt to find the roots of this polynomial using techniques such as synthetic division or the Rational Root Theorem, or we can use numerical methods or a graphing approach to find approximate solutions.
To find rational roots, we can check possible rational roots such as \( x = 1 \) and \( x = -1 \):
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Testing \( x = 1 \): \[ 1^3 - 5(1^2) - 1 + 5 = 1 - 5 - 1 + 5 = 0 \] So \( x = 1 \) is a root.
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Testing \( x = -1 \): \[ (-1)^3 - 5(-1)^2 - (-1) + 5 = -1 - 5 + 1 + 5 = 0 \] So \( x = -1 \) is also a root.
Now we can factor the cubic polynomial using the roots we found, starting with \( x = 1 \) and \( x = -1 \). Using synthetic division to factor out \( (x - 1) \) and \( (x + 1) \):
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Dividing by \( (x - 1) \) gives us: \[ x^3 - 5x^2 - x + 5 = (x - 1)(x^2 - 4x - 5) \]
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Next, we can factor \( x^2 - 4x - 5 \) as: \[ x^2 - 4x - 5 = (x - 5)(x + 1) \]
So the full factorization of the polynomial is:
\[ f(x) - g(x) = (x - 1)(x + 1)(x - 5) \]
From this factorization, the roots are \( x = 1 \), \( x = -1 \), and \( x = 5 \).
Now we analyze the meaning of these solutions:
- The solutions \( x = -1 \), \( x = 1 \), and \( x = 5 \) signify the \( x \)-values where the graphs of the two functions \( f(x) \) and \( g(x) \) intersect. Therefore, this means that \( f(x) \) and \( g(x) \) have three points of intersection.
Now, we can determine which of the statements is correct based on our findings:
- The correct interpretation is:
- 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.
Thus, the correct response from the provided options is:
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.