Problem 58 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find all the real zeros of the polynomial function. Determine the multiplicity of each zero. Use a graphing utility to verify your results. \(f(x)=\frac{5}{3} x^{2}+\frac{8}{3} x-\frac{4}{3}\)

Step-by-Step Solution

Answer

The zeros of the quadratic polynomial function \( f(x) = \frac{5}{3} x^{2} + \frac{8}{3} x - \frac{4}{3} \) are \( x = - \frac{8}{10} + \frac{2\sqrt{21}}{5} \) and \( x = - \frac{8}{10} - \frac{2\sqrt{21}}{5} \), each with multiplicities of one.

Step 1: Understanding the given polynomial and applying the quadratic formula

Given the quadratic polynomial \( f(x)=\frac{5}{3} x^{2}+\frac{8}{3} x-\frac{4}{3}\). The quadratic formula is used to find the roots of the quadratic equation and is given by: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) where a, b, and c are coefficients of the quadratic equation \( ax^2 + bx + c = 0 \). Consequently, the calculation will proceed by substituting a=\(\frac{5}{3}\), b=\(\frac{8}{3}\) and c=-\(\frac{4}{3}\) into the formula.

Step 2: Calculation of the roots

Substitute the values of a, b, and c into the quadratic formula. That is, \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-\frac{8}{3} \pm \sqrt{(\frac{8}{3})^2 - 4*\frac{5}{3}*(-\frac{4}{3})}}{2*\frac{5}{3}} \). Evaluating within the square root we get \( x = \frac{-\frac{8}{3} \pm \sqrt{ (\frac{64}{9}) + (\frac{20}{9})}}{2*\frac{5}{3}} = \frac{-\frac{8}{3} \pm \sqrt{\frac{84}{9}}}{\frac{10}{3}} \). The square root of \(\frac{84}{9}\) is \(\frac{4\sqrt{21}}{3}\). So, the roots become \( x = \frac{-8 \pm 4\sqrt{21}}{10} \), i.e., the roots are \( x = - \frac{8}{10} + \frac{2\sqrt{21}}{5} \) and \( x = - \frac{8}{10} - \frac{2\sqrt{21}}{5} \)

Step 3: Determination of the multiplicities of the roots

The polynomial is a quadratic equation of degree 2 and we have found two different roots for the equation. Hence, the multiplicity of each root is one. Even without simplifying the roots, one clearly sees that the + and - cases lead to two distinct real numbers.

Step 4: Graphing the function to verify results

To confirm if the calculated roots and their multiplicities are correct, plot the function \( f(x)=\frac{5}{3} x^{2}+\frac{8}{3} x-\frac{4}{3} \) using a graphing utility. If the results are correct, the graph of the function should touch the x-axis at each of the roots and the behavior of the graph at the roots should align with their multiplicities. For example, the graph goes through at roots with multiplicity one. If it looks like this, the solutions found in the previous steps are correct.

Key Concepts

Real Zeros

Real zeros of a quadratic equation are the x-values where the graph of the equation touches or crosses the x-axis. These are also known as the roots or solutions of the quadratic equation. In our case, the quadratic formula helps us find the real zeros of the function \(f(x)=\frac{5}{3} x^{2}+\frac{8}{3} x-\frac{4}{3}\). Using this formula, we can determine the two real zeros by solving:

• \( x = \frac{-b + \sqrt{b^2 - 4ac}}{2a} \)
• \( x = \frac{-b - \sqrt{b^2 - 4ac}}{2a} \)

These are the values of \(x\) that make the function equal to zero. By substituting values \(a=\frac{5}{3}\), \(b=\frac{8}{3}\), \(c=-\frac{4}{3}\), we calculate the zeros which, in this scenario, are real and distinct due to the positive value under the square root, \(b^2 - 4ac\). This value is often called the discriminant. A positive discriminant indicates two distinct real zeros.

Multiplicity of Roots

The multiplicity of a root tells us how a quadratic function behaves at its roots on the graph. Specifically, it indicates how many times a particular zero occurs. For quadratic equations, each root can have a multiplicity of 1 or 2:

• A root with multiplicity 1 means the graph crosses the x-axis at this point.
• A root with multiplicity 2 means the graph touches the x-axis without crossing it at the root, resembling a tangent line.

In this exercise, since we have two distinct real zeros found through the quadratic formula, each root has a multiplicity of 1. This indicates that the graph will cross the x-axis at each of these points. Generally, the sum of the multiplicities of the roots will equal the degree of the polynomial. Because this is a quadratic (degree 2), the sum of the multiplicities is also 2, confirming our findings.

Graphing Quadratic Functions

Graphing a quadratic function is an effective way to visually confirm the real zeros and their multiplicities. A quadratic function can be graphed as a parabola which opens upwards if \(a\) is positive and downwards if \(a\) is negative. For the given function \(f(x)=\frac{5}{3} x^{2}+\frac{8}{3} x-\frac{4}{3}\):

• The graph is a parabola that opens upwards since \(a=\frac{5}{3}\) is positive.
• The x-intercepts on the graph correspond to the real zeros previously calculated.
• Due to the roots' multiplicity of 1, the graph crosses the x-axis at these points.

By plotting the function using a graphing tool, students can see how the parabola intersects the x-axis at the calculated roots. This visual approach not only confirms the solutions but also aids in understanding the quadratic nature of the equation and its graph.