Answer :
To solve for the function [tex]\( f(x) \)[/tex], we start by analyzing its given polynomial expression:
[tex]\[ f(x) = 5x^4 - 26x^3 - 66x^2 + 19x + 14 \][/tex]
This is a fourth-degree polynomial, which implies that its general form can accommodate up to four real roots. However, solving for roots analytically could be quite complex due to its higher degree, and often numerical methods or graphing are employed in practical applications.
Without diving into root-finding procedures directly, let's consider the key characteristics of this polynomial function:
### Leading Term Analysis:
The leading term of the polynomial is [tex]\( 5x^4 \)[/tex]. The coefficient [tex]\( 5 \)[/tex] is positive, and since the degree is even (4), this means as [tex]\( x \)[/tex] approaches [tex]\( \pm \infty \)[/tex], [tex]\( f(x) \)[/tex] will tend to [tex]\( +\infty \)[/tex]. This behavior informs us that the polynomial will open upwards on both ends of the graph.
### Derivative and Critical Points:
To find the critical points (where the function could have local maxima or minima), you would typically take the first derivative [tex]\( f'(x) \)[/tex] and solve for the points where [tex]\( f'(x) = 0 \)[/tex].
However, let's focus on the given function formulation, which directly evaluates the function without going into derivatives:
Given the polynomial:
[tex]\[ f(x) = 5x^4 - 26x^3 - 66x^2 + 19x + 14 \][/tex]
This function can be evaluated at any particular value of [tex]\( x \)[/tex] to find the corresponding [tex]\( f(x) \)[/tex]. For instance:
1. Evaluate [tex]\( f(0) \)[/tex]:
[tex]\[ f(0) = 5(0)^4 - 26(0)^3 - 66(0)^2 + 19(0) + 14 = 14 \][/tex]
2. Evaluate [tex]\( f(1) \)[/tex]:
[tex]\[ f(1) = 5(1)^4 - 26(1)^3 - 66(1)^2 + 19(1) + 14 = 5 - 26 - 66 + 19 + 14 = -54 \][/tex]
3. Evaluate [tex]\( f(-1) \)[/tex]:
[tex]\[ f(-1) = 5(-1)^4 - 26(-1)^3 - 66(-1)^2 + 19(-1) + 14 = 5 + 26 - 66 - 19 + 14 = -40 \][/tex]
### Summary:
The polynomial function [tex]\( f(x) = 5x^4 - 26x^3 - 66x^2 + 19x + 14 \)[/tex] exhibits behavior typical of a fourth-degree polynomial with positive leading coefficient: it rises to positive infinity as [tex]\( x \)[/tex] moves to positive or negative infinity. We can evaluate this polynomial at specific points to understand its value distribution, as done in the examples.
This should cover a fundamental understanding as to how the polynomial can be evaluated and some of its general characteristics. For more detailed analysis, numerical or graphing tools are often used.
[tex]\[ f(x) = 5x^4 - 26x^3 - 66x^2 + 19x + 14 \][/tex]
This is a fourth-degree polynomial, which implies that its general form can accommodate up to four real roots. However, solving for roots analytically could be quite complex due to its higher degree, and often numerical methods or graphing are employed in practical applications.
Without diving into root-finding procedures directly, let's consider the key characteristics of this polynomial function:
### Leading Term Analysis:
The leading term of the polynomial is [tex]\( 5x^4 \)[/tex]. The coefficient [tex]\( 5 \)[/tex] is positive, and since the degree is even (4), this means as [tex]\( x \)[/tex] approaches [tex]\( \pm \infty \)[/tex], [tex]\( f(x) \)[/tex] will tend to [tex]\( +\infty \)[/tex]. This behavior informs us that the polynomial will open upwards on both ends of the graph.
### Derivative and Critical Points:
To find the critical points (where the function could have local maxima or minima), you would typically take the first derivative [tex]\( f'(x) \)[/tex] and solve for the points where [tex]\( f'(x) = 0 \)[/tex].
However, let's focus on the given function formulation, which directly evaluates the function without going into derivatives:
Given the polynomial:
[tex]\[ f(x) = 5x^4 - 26x^3 - 66x^2 + 19x + 14 \][/tex]
This function can be evaluated at any particular value of [tex]\( x \)[/tex] to find the corresponding [tex]\( f(x) \)[/tex]. For instance:
1. Evaluate [tex]\( f(0) \)[/tex]:
[tex]\[ f(0) = 5(0)^4 - 26(0)^3 - 66(0)^2 + 19(0) + 14 = 14 \][/tex]
2. Evaluate [tex]\( f(1) \)[/tex]:
[tex]\[ f(1) = 5(1)^4 - 26(1)^3 - 66(1)^2 + 19(1) + 14 = 5 - 26 - 66 + 19 + 14 = -54 \][/tex]
3. Evaluate [tex]\( f(-1) \)[/tex]:
[tex]\[ f(-1) = 5(-1)^4 - 26(-1)^3 - 66(-1)^2 + 19(-1) + 14 = 5 + 26 - 66 - 19 + 14 = -40 \][/tex]
### Summary:
The polynomial function [tex]\( f(x) = 5x^4 - 26x^3 - 66x^2 + 19x + 14 \)[/tex] exhibits behavior typical of a fourth-degree polynomial with positive leading coefficient: it rises to positive infinity as [tex]\( x \)[/tex] moves to positive or negative infinity. We can evaluate this polynomial at specific points to understand its value distribution, as done in the examples.
This should cover a fundamental understanding as to how the polynomial can be evaluated and some of its general characteristics. For more detailed analysis, numerical or graphing tools are often used.
