Analyzing And Graphing A Fourth-Degree Polynomial Function

f(x)=(x^4)+7x^2+5

f(x)=12x^2+14

The expression you’ve given is a fourth-degree polynomial function, f(x) = x^4 + 7x^2 + 5. Here are some explanations and strategies you can use to analyze, graph, and simplify this function:

1. Degree: The degree of a polynomial function is the highest exponent on the variable. In this case, the degree is 4, which means the function is a quartic (fourth-degree) polynomial.

2. Coefficients: The coefficients multiply each term in the polynomial. In this case, the coefficients are 1 (the coefficient of x^4), 7 (the coefficient of x^2), and 5 (the constant term).

3. Roots: The roots (or zeros) of a function are the values of x for which the function equals zero. To find the roots of a polynomial function, you can factor it, use the quadratic formula (for quadratic functions), or use numerical methods (like a graphing calculator). For this function, f(x) = x^4 + 7x^2 + 5, there are no real roots because the discriminant (b^2 – 4ac) of the quadratic equation that results from setting f(x) = 0 is negative.

4. Graph: To graph the function, you can use a graphing calculator, plot some points by hand, or analyze the behavior of the function. Since the function is a fourth-degree polynomial, it has the same end behavior as y = x^4 – a (for some value of a). This means that as x approaches positive or negative infinity, the function increases without bound if a < 0, or decreases without bound if a > 0. To find the relative extrema of the function (where it has maximum or minimum values), you can take the derivative and set it equal to zero. In this case, the derivative is f'(x) = 4x^3 + 14x, which factors as 2x(2x^2 + 7) = 0. The critical values are x = 0 and x = sqrt(7/2) or x = -sqrt(7/2). A little analysis shows that f is increasing on (-inf, -sqrt(7/2)) and (0, sqrt(7/2)) and decreasing on (-sqrt(7/2), 0) and (sqrt(7/2), inf). Therefore, f has a local minimum at x = -sqrt(7/2) and a local maximum at x = sqrt(7/2) (because the power of x is even).

5. Simplification: To simplify the function, you can try factoring it or using algebraic techniques (like completing the square). However, this function doesn’t factor nicely, so there isn’t a simple way to rewrite it. One possible technique is to substitute a new variable (like u = x^2) to obtain a simpler expression, but this doesn’t always work. In general, you can simplify a polynomial function by identifying patterns (like symmetry, end behavior, or common factors) or by relating it to other functions (like exponentials, trig functions, or logarithms).

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