Problem 57 - 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. \(f(x)=\frac{1}{2} x^{2}+\frac{5}{2} x-\frac{3}{2}\)

Step-by-Step Solution

Answer

The roots of \(f(x)=\frac{1}{2} x^{2}+\frac{5}{2} x-\frac{3}{2}\) are \(x = 2\) and \(x = 3\) each with multiplicity 1.

Step 1: Apply the Quadratic Formula

The quadratic formula \(x =\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) is applied to find the roots of the polynomial. Given the function \(f(x)=\frac{1}{2} x^{2}+\frac{5}{2} x-\frac{3}{2}\), where \(a = \frac{1}{2}\), \(b = \frac{5}{2}\), and \(c = -\frac{3}{2}\), the formula gives \(x=\frac{-\frac{5}{2} \pm \sqrt{(\frac{5}{2})^2 - 4*\frac{1}{2}*(-\frac{3}{2})}}{2*\frac{1}{2}}\).

Step 2: Calculate Roots

Simplify the expression to get the roots. The roots will be \(x=\frac{-\frac{5}{2} \pm \frac{1}{2}}{\frac{1}{2}}\). There are two roots given by \(x = 2\) and \(x = 3\).

Step 3: Determine the Multiplicity of Each Root

Each solution of the quadratic formula produces a root of the polynomial. Solutions are both different, so their respective multiplicities are both 1.

Step 4: Verification using a Graphing Utility

Using a graphing utility, plot the function \(f(x)=\frac{1}{2} x^{2}+\frac{5}{2} x-\frac{3}{2}\) . The graph will intersect the x-axis at \(x = 2\) and \(x = 3\), corresponding to the roots found, which confirms the solution.

Key Concepts

Real Zeros

Real zeros of a polynomial function are the points where the graph of the function intersects the x-axis. To put it simply, these are the values of \(x\) that make the function equal to zero. In this problem, after applying the quadratic formula, we find that the real zeros of the function \(f(x)=\frac{1}{2} x^{2}+\frac{5}{2} x-\frac{3}{2}\) are \(x = 2\) and \(x = 3\). These zeros are important because they represent the solutions to the equation \(f(x) = 0\). To verify the real zeros, you can graph the polynomial equation. When you plot the graph, observe the 'x' points where the plot touches or crosses the x-axis. If the graph intersects at 2 and 3, it matches your calculations.

Multiplicity of Roots

The multiplicity of a root refers to the number of times a particular root is repeated in a polynomial equation. In simple terms, it indicates how many times a particular solution, or zero, appears in the set of solutions. It is crucial because it affects the behavior of the graph at the zero.

• If a root has an odd multiplicity, the graph will cross the x-axis at this root.
• If a root has an even multiplicity, the graph will touch the x-axis but not cross it.

For this problem, after finding the zeros \(x = 2\) and \(x = 3\), each zero has a multiplicity of 1. This means each root appears once, and the graph crosses the x-axis at these points.

Quadratic Formula

The quadratic formula is a powerful tool for finding the real roots of any quadratic equation. The standard form of a quadratic equation is \(ax^2 + bx + c = 0\). The quadratic formula is given by:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Let's break this down further:

• \(-b\): This is the negative of the coefficient \(b\), which shifts the vertex of the parabola along the x-axis.
• \(\pm\): This symbol indicates that there are generally two solutions: one adding \(\sqrt{b^2 - 4ac}\), and one subtracting it.
• \(\sqrt{b^2 - 4ac}\): Known as the discriminant, it determines the nature of the roots. If it is positive, there are two distinct real roots; if zero, there's exactly one real root (a repeated root); and if negative, the roots are complex and not real.
• \(2a\): The solution divides by \(2a\), normalizing the output for the coefficient of \(x^2\).

In this case, applying the formula to the given polynomial provides the real zeros \(x = 2\) and \(x = 3\). Utilizing the quadratic formula ensures that you can systematically find the zeros of any quadratic function, no matter how complex it may seem.