Given the functions f(x)=x3+5

Question

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)
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.

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=−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)
each have three x-intercepts.

Answer 1

To solve the equation \( f(x) = g(x) \) where \( f(x) = x^3 + 5 \) and \( g(x) = 5x^2 + x \), we need to equate the two functions:

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

Rearranging this equation gives:

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

Now, to find the roots of this cubic equation, we can use numerical methods, factorization, or graphing. For the sake of understanding, let's check if we can find rational roots through substitution or the Rational Root Theorem.

Testing a few possible integer roots:

1. Testing \( x = 1 \): \[ 1^3 - 5(1^2) - 1 + 5 = 1 - 5 - 1 + 5 = 0 \] Thus, \( x = 1 \) is a root.

Now we can perform synthetic division or polynomial division to factor \( x - 1 \) out of the cubic polynomial:

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 need to factor \( x^2 - 4x - 5 \):

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

Putting it all together, we have:

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

So the roots are:

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

These roots mean that the equation \( f(x) = g(x) \) has three solutions: \( x = 1 \), \( x = 5 \), and \( x = -1 \).

Interpretation of Solutions

These solutions indicate that the graphs of the functions \( f(x) \) and \( g(x) \) intersect at three points. The \( x \)-intercepts of each function provide information about their graphical representation, but the important note here is that the three points of intersection mean that \( f(x) \) and \( g(x) \) share these intersection points.

Correct Statement

Given our findings, the correct statement from the options provided 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."