Problem 71 - Algebra for College Students

Question

Find all real roots to each polynomial equation by graphing the corresponding function and locating the x-intercepts. $$x^{4}-12 x^{2}+10=0$$

Step-by-Step Solution

Answer

The real roots of the equation \(x^{4}-12 x^{2}+10=0\) are the x-intercepts found on the graph of \(f(x)=x^{4} - 12 x^{2} + 10\).

Step 1: Identify the polynomial equation

The given polynomial equation is \(x^{4}-12 x^{2}+10=0\).

Step 2: Rewrite the equation to identify the related function

Consider the function \(f(x) = x^{4} - 12x^{2} + 10\). To find the real roots of the equation, we need to determine the x-intercepts of this function.

Step 3: Graph the function

Use a graphing calculator or software to graph the function \(f(x) = x^{4} - 12x^{2} + 10\). Carefully analyze the graph to locate the points where the graph intersects the x-axis.

Step 4: Identify the x-intercepts of the graph

Observe the x-intercepts on the graph. These are the values of \(x\) where \(f(x)=0\). The x-intercepts correspond to the real roots of the polynomial equation.

Step 5: List the real roots

From the graph, identify the specific x-values where the graph intersects the x-axis. These are the roots of the equation.

Key Concepts

polynomial equation

A polynomial equation is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. For instance, the given polynomial equation is \(x^{4}-12 x^{2}+10=0\). This particular equation is a fourth-degree polynomial because the highest power of \(x\) in the equation is 4. Understanding polynomial equations is crucial because they model a wide range of real-world phenomena, from physics to economics.
In this example, solving the polynomial equation means finding the values of \(x\) that make the equation equal to zero. These values are called the 'roots' of the equation.

real roots

Real roots of a polynomial equation are the values of \(x\) that satisfy the equation and are real numbers (not imaginary). In simple terms, when you plug these values back into the equation, the left-hand side becomes zero. For example, for the polynomial \(x^{4}-12 x^{2}+10=0\), the real roots are the values that make this equation true.
Identifying real roots is essential because these roots are the points where the polynomial crosses or touches the x-axis on a graph. Graphing is one effective way to find the real roots, as it visually represents where the polynomial equals zero.

graphing functions

Graphing functions is a fundamental technique in mathematics used to understand the behavior of different types of functions, including polynomial equations. When you graph a polynomial function like \(f(x) = x^{4} - 12x^{2} + 10\), you can easily see where the function intersects the x-axis. These intersection points correlate with the real roots of the equation.
To graph the function, you can use a graphing calculator or software. This visual tool helps you plot various values of \(x\) and their corresponding \(f(x)\) values to draw the curve, clearly showing the critical points, including maximums, minimums, and the x-intercepts.

x-intercepts

The x-intercepts of a function are the points where the graph of the function crosses the x-axis. These points represent the real roots of the polynomial equation because the function value \(f(x)\) is zero at these points. In other words, if \(f(x) = x^{4} - 12x^{2} + 10\), the x-intercepts are the values of \(x\) where \(f(x) = 0\).
Finding x-intercepts involves graphing the function and observing where it hits the x-axis. For our polynomial, once we graph it, we look for the x-values where the graph crosses the x-axis. These x-values are the roots of the given polynomial equation.