Question
Given the functions f(x)=x3+5 and g(x)=5x2+x , solve the equation f(x)=g(x) and interpret the solutions. Which of the following statements is correct?(1 point) Responses The solution to f(x)=g(x) is x=−54 and x=1 . This means that f(x) and g(x) have two points of intersection. The solution to f left parenthesis x right parenthesis equals g left parenthesis x right parenthesis is x equals negative Start Fraction 5 over 4 End Fraction and x equals 1 . This means that f left parenthesis x right parenthesis and g left parenthesis x right parenthesis have two points of intersection. The solution to 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. The solution to f left parenthesis x right parenthesis equals g left parenthesis x right parenthesis is x equals negative 1 , x equals 1 , and x equals 5 . This means that f left parenthesis x right parenthesis and g left parenthesis x right parenthesis have three points of intersection. 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. The solution to f left parenthesis x right parenthesis equals g left parenthesis x right parenthesis is x equals negative 1 , x equals 1 , and x equals 5 . This means that f left parenthesis x right parenthesis and g left parenthesis x right parenthesis each have three x -intercepts. The solution to f(x)=g(x) is x=−54 and x=1 . This means that f(x) and g(x) each have two x-intercepts. The solution to f left parenthesis x right parenthesis equals g left parenthesis x right parenthesis is x equals negative Start Fraction 5 over 4 End Fraction and x equals 1 . This means that f left parenthesis x right parenthesis and g left parenthesis x right parenthesis each have two x -intercepts.
Answers
GPT-4o mini
To solve the equation \( f(x) = g(x) \) given the functions:
\[
f(x) = x^3 + 5
\]
\[
g(x) = 5x^2 + x
\]
we set these two functions equal to each other:
\[
x^3 + 5 = 5x^2 + x
\]
Rearranging the equation gives:
\[
x^3 - 5x^2 - x + 5 = 0
\]
To find the roots of the polynomial \( x^3 - 5x^2 - x + 5 = 0 \), we can use methods such as synthetic division, the Rational Root Theorem, or numerical root-finding methods. First, it's helpful to test for possible rational roots (values of \( x \)).
After testing some reasonable values, we find that \( x = 1 \) is a root:
\[
1^3 - 5(1^2) - 1 + 5 = 1 - 5 - 1 + 5 = 0
\]
Now, we can perform synthetic division or polynomial long division on \( x^3 - 5x^2 - x + 5 \) using \( x - 1 \):
Dividing \( x^3 - 5x^2 - x + 5 \) by \( x - 1 \) gives:
\[
x^3 - 5x^2 - x + 5 = (x - 1)(x^2 - 4x - 5)
\]
Next, we factor \( x^2 - 4x - 5 \):
\[
x^2 - 4x - 5 = (x - 5)(x + 1)
\]
Thus, we can write:
\[
x^3 - 5x^2 - x + 5 = (x - 1)(x - 5)(x + 1)
\]
Setting each factor to zero gives the solutions:
1. \( x - 1 = 0 \) → \( x = 1 \)
2. \( x - 5 = 0 \) → \( x = 5 \)
3. \( x + 1 = 0 \) → \( x = -1 \)
So, the solutions are:
\[
x = -1, \; x = 1, \; x = 5
\]
**Interpretation:**
These solutions mean that the graphs of \( f(x) \) and \( g(x) \) intersect at three points: \( (-1, f(-1)) \), \( (1, f(1)) \), and \( (5, f(5)) \).
Let's interpret the correct statement:
"The solution to \( f(x) = g(x) \) is \( x = -1, x = 1, \) and \( x = 5 \). This means that \( f(x) \) and \( g(x) \) have three points of intersection."
Thus, the correct statement from the provided options is:
**"The solution to 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."**
\[
f(x) = x^3 + 5
\]
\[
g(x) = 5x^2 + x
\]
we set these two functions equal to each other:
\[
x^3 + 5 = 5x^2 + x
\]
Rearranging the equation gives:
\[
x^3 - 5x^2 - x + 5 = 0
\]
To find the roots of the polynomial \( x^3 - 5x^2 - x + 5 = 0 \), we can use methods such as synthetic division, the Rational Root Theorem, or numerical root-finding methods. First, it's helpful to test for possible rational roots (values of \( x \)).
After testing some reasonable values, we find that \( x = 1 \) is a root:
\[
1^3 - 5(1^2) - 1 + 5 = 1 - 5 - 1 + 5 = 0
\]
Now, we can perform synthetic division or polynomial long division on \( x^3 - 5x^2 - x + 5 \) using \( x - 1 \):
Dividing \( x^3 - 5x^2 - x + 5 \) by \( x - 1 \) gives:
\[
x^3 - 5x^2 - x + 5 = (x - 1)(x^2 - 4x - 5)
\]
Next, we factor \( x^2 - 4x - 5 \):
\[
x^2 - 4x - 5 = (x - 5)(x + 1)
\]
Thus, we can write:
\[
x^3 - 5x^2 - x + 5 = (x - 1)(x - 5)(x + 1)
\]
Setting each factor to zero gives the solutions:
1. \( x - 1 = 0 \) → \( x = 1 \)
2. \( x - 5 = 0 \) → \( x = 5 \)
3. \( x + 1 = 0 \) → \( x = -1 \)
So, the solutions are:
\[
x = -1, \; x = 1, \; x = 5
\]
**Interpretation:**
These solutions mean that the graphs of \( f(x) \) and \( g(x) \) intersect at three points: \( (-1, f(-1)) \), \( (1, f(1)) \), and \( (5, f(5)) \).
Let's interpret the correct statement:
"The solution to \( f(x) = g(x) \) is \( x = -1, x = 1, \) and \( x = 5 \). This means that \( f(x) \) and \( g(x) \) have three points of intersection."
Thus, the correct statement from the provided options is:
**"The solution to 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."**