Problem 43 - College Algebra and Calculus: An Applied Approach

Question

Algebraic and Graphical Approaches In Exercises \(31-46\), find all real zeros of the function algebraically. Then use a graphing utility to confirm your results. $$f(x)=2 x^{4}-2 x^{2}-40$$

Step-by-Step Solution

Answer

The real zeros of the function \(f(x)=2x^{4}-2x^{2}-40\) are \(\sqrt{5}\) and \(-\sqrt{5}\).

Step 1: Factorizing the Polynomial

The polynomial \(f(x)=2x^{4}-2x^{2}-40\) can be rewritten as \(2x^{4}-2x^{2}=40\). For simplicity, it's more convenient to consider a substitution \(y = x^{2}\), which transforms the polynomial into the quadratic equation in the form \(2y^2 - 2y - 40 = 0\). This equation can be further simplified by dividing by '2' to \(y^{2}-y-20=0\).

Step 2: Finding Real Zeros Algebraically

The quadratic equation (from Step 1, \(y^{2}-y-20=0\)) can be factored into \((y-5)(y+4)=0\). Setting the factors equal to zero and solving for \(y\) gives \(y=5\) and \(y=-4\). Remember that we made a substitution \(y=x^{2}\) at the beginning. So, we get \(x^{2}=5\) and \(x^{2}=-4\). The first equation gives \(x=\sqrt{5}\) and \(x=-\sqrt{5}\) as real zeros. The second equation, \(x^{2}=-4\), has no real solutions because the square of a real number is always positive or zero. Hence, the real zeros of the function \(f(x)\) are \(\sqrt{5}\) and \(-\sqrt{5}\).

Step 3: Verifying the Real Zeros Using a Graphing Utility

To confirm the solutions graphically, a graphing utility can be used to plot the function \(f(x)=2x^{4}-2x^{2}-40\). The x-coordinates of the points where the curve crosses or touches the x-axis represent the real zeros of the function. When the function is graphed, it can be seen that the graph intersects the x-axis at the points \(\sqrt{5}\) and \(-\sqrt{5}\), thereby verifying the solutions. Always note, the precise intersection points might depend on the accuracy of the graphing utility.

Key Concepts

Polynomial Function

A polynomial function is an expression consisting of variables raised to whole number exponents and coefficients. For example, the given function, \( f(x) = 2x^4 - 2x^2 - 40 \), is a polynomial of degree 4, because the highest exponent is 4.
Polyomial functions can have one or more terms, and each term is composed of a coefficient, a variable, and an exponent.
Understanding the structure of a polynomial is crucial, as it helps in recognizing the possible number of roots, or zeros, the function could have. This understanding is the first step in solving polynomial equations.

Real Zeros

Real zeros of a polynomial function are the values of \( x \) which make the function equal to zero. In the equation \( f(x) = 0 \), real zeros are the solutions to this equation.

• They represent the x-values where the graph of the polynomial touches or crosses the x-axis.
• Finding real zeros algebraically often involves factoring the polynomial or using the quadratic formula for simpler equations.

In our example, we found the real zeros by setting \( y = x^2 \) and then solving \( y^2 - y - 20 = 0 \). The solutions \( y = 5 \) lead to the real zeros \( x = \sqrt{5} \) and \( x = -\sqrt{5} \). These indicate the intersection points on the graph.

Factoring

Factoring transforms polynomials into a product of simpler expressions, which can make solving for zeros easier. In the given problem, the polynomial was rewritten using a substitution variable \( y = x^2 \).
This turned the problem into a simpler quadratic equation, \( y^2 - y - 20 = 0 \).
Factoring this quadratic into \( (y-5)(y+4) = 0 \) shows how the original complex polynomial can be handled through factoring techniques.

• By setting each factor equal to zero, we found the "zero points", which are the values that satisfy the original polynomial equation.

Graphing Utility

A graphing utility is a useful tool for visualizing polynomial functions and verifying solutions found algebraically. By plotting the function \( f(x) = 2x^4 - 2x^2 - 40 \), you can observe where the graph intersects the x-axis.
These positions correspond to the real zeros of the polynomial and provide a visual confirmation of algebraic work.
However, it's essential to remember:

• The accuracy of identifying zeros depends on the resolution and scale of the graph.
• Slight errors in plotting might occur with graphing utilities, but they are invaluable for checking solutions.

Using a graphing utility helps solidify understanding of abstract algebraic results through concrete graphical evidence.