Problem 54 - 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. \(f(x)=2 x^{2}-14 x+24\)

Step-by-Step Solution

Answer

The zeros of the polynomial are \(x = 4\) and \(x = 3\), each with a multiplicity of 1.

Step 1: Set the Function Equal to Zero

In order to find the zeros, set the function \(f(x) = 2x^{2} - 14x + 24\) equal to zero, like so: \(2x^{2} - 14x + 24 = 0\).

Step 2: Solve for x

To solve this quadratic equation, use the quadratic formula \(x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\). In the given equation, \(a = 2\), \(b = -14\), and \(c = 24\). \(x = \frac{-(-14) \pm \sqrt{(-14)^{2} - 4 \cdot 2 \cdot 24}}{2\cdot 2}\). After calculating, we get \(x_{1}=4\), \(x_{2}=3\).

Step 3: Find the Multiplicity

The given polynomial can be factored as \([2(x-4)(x-3)]\), so both 4 and 3 appear as roots of the polynomial only once. Therefore, the multiplicity of each root is 1.

Step 4: Verify Using a Graphing Utility

Plotting the function using a graphing utility, it can be seen that the graph of the polynomial intersects the x-axis at x=3 and x=4, confirming the earlier calculation.

Key Concepts

Quadratic Equation

A quadratic equation is a polynomial equation of degree two, meaning the highest exponent of the variable, usually denoted as \(x\), is two. The standard form of a quadratic equation is \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. In this case, \(a\) should not be equal to zero, because if \(a=0\), then the equation would not be quadratic. Quadratic equations can have different types of solutions, including:

• Two distinct real solutions
• One real solution (a repeated root)
• No real solutions (but complex solutions)

The solutions are also known as the "zeros" of the quadratic function \(f(x) = ax^2 + bx + c\). They represent the points where the graph of the function intersects the x-axis. Solving quadratic equations can be achieved through multiple methods, including factoring, completing the square, and using the quadratic formula.

Quadratic Formula

The quadratic formula is a powerful tool for finding the zeros of a quadratic equation. It is derived from the process of completing the square and is expressed as \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). This formula can be used whether or not the quadratic equation is easily factorable.

The expression under the square root sign, \(b^2 - 4ac\), is referred to as the discriminant. It provides crucial information about the nature of the roots of the quadratic equation:

• If \(b^2 - 4ac > 0\), there are two distinct real roots.
• If \(b^2 - 4ac = 0\), there is one real root (or a repeated root).
• If \(b^2 - 4ac < 0\), the roots are complex and not real.

In the exercise provided, the quadratic formula was used to determine the zeros \(x_1 = 4\) and \(x_2 = 3\) by substituting \(a = 2\), \(b = -14\), and \(c = 24\) into the formula.

Multiplicity of Zeros

Multiplicity refers to the number of times a particular zero appears as a root of a polynomial equation. In simple terms, it's how many times a factor appears in the factorized form of the polynomial. Zeros can have different multiplicities:

• A zero with a multiplicity of 1 is called a "simple zero" and means the graph of the polynomial crosses the x-axis at this point.
• A zero with a higher multiplicity typically means the graph touches the x-axis at this point but does not cross it. For instance, a zero of multiplicity 2 would appear as a "peak" or "valley" on the graph.

In the given polynomial \(f(x)=2x^2-14x+24\), factoring the polynomial as \(2(x-4)(x-3)\) reveals that both zeros \(x=4\) and \(x=3\) have a multiplicity of 1. This implies that the graph crosses the x-axis at these values, as confirmed by a graphing utility in the step-by-step solution.