Problem 31 - Algebra and Trigonometry

Question

In Exercises \(25-32,\) find the zeros for each polynomial function and give the multiplicity for each zero. State whether the graph crosses the \(x\) -axis, or touches the \(x\) -axis and turns around, at each zero. $$f(x)=x^{3}+7 x^{2}-4 x-28$$

Step-by-Step Solution

Answer

The zeros of the function \(f(x)=x^{3}+7 x^{2}-4 x-28\) are \(x=0\), \(x=-4\), and \(x=-7\), each with a multiplicity of \(1\). The graph crosses the \(x\)-axis at each zero.

Step 1: Finding the zeros

To find the zeros of the polynomial \(f(x)=x^{3}+7 x^{2}-4 x-28\), we set the function equal to zero and solve for \(x:\) \(0=x^{3}+7 x^{2}-4 x-28\). Factoring the polynomial, we get: \(x(x+4)(x+7)=0\). Setting each factor equal to zero, the solutions are \(x=0\), \(x=-4\), \(x=-7\)

Step 2: Determining Multiplicity of each zero and checking the behavior of the graph

The zeros of the function are \(0\), \(-4\), and \(-7\). Since each of these zeros is produced from different factors, each of them appears only once, and therefore, each has a multiplicity of \(1\). Because all multiplicities are odd, the graph crosses the \(x\)-axis at each zero.

Key Concepts

Understanding Polynomial Zeros and Multiplicity

The zeros of a polynomial function are the values of x that make the function equal to zero. These are crucial as they are the points where the graph of the function intersects the x-axis. Significantly, each zero has an associated multiplicity, which refers to the number of times a particular zero appears as a root in the polynomial equation. For instance, if we factor a polynomial and find that (x-2) appears twice in the product, the zero at x=2 would have a multiplicity of 2.

Multiplicity affects the graph's behavior at the x-axis. If a zero has an odd multiplicity, the graph of the function will cross the x-axis at that point. Conversely, with an even multiplicity, the graph merely touches the x-axis and turns around at the zero. In the given exercise, the polynomial function f(x) = x^3 + 7x^2 - 4x - 28 has zeros with a multiplicity of 1, meaning the graph will intersect the x-axis at each zero, providing a visual understanding of the function's behavior in a graphical representation.

Graph Behavior at Zeros

The behavior of a polynomial graph around its zeros gives insight into the function's nature. When dealing with multiplicity, it's important to know that the curve will behave differently at each point where it hits or touches the x-axis. As mentioned, zeros with odd multiplicities will result in the graph crossing the axis. Imagine it as a car driving over a hill – it goes up one side and down the other. On the other hand, zeros with even multiplicities show the graph approaching the axis and taking a U-turn, much like a car reaching the top of a hill and reversing direction.

The exercise provided illustrates that for f(x) = x^3 + 7x^2 - 4x - 28, because each zero has a multiplicity of 1 (which is odd), the graph crosses the x-axis at the points x=0, x=-4, and x=-7. This behavior can often help us predict the shape and curves of the graph even without plotting it.

Factoring Polynomials

Factoring polynomials is a method we use to break down a polynomial into simpler components that are easier to understand and solve. This method is akin to breaking down a complex problem into smaller, more manageable parts. When we successfully factor a polynomial, we can easily identify its zeros. In fact, each factor of the form (x - c), where c is a constant number, represents a zero of the polynomial at x=c.

Returning to our exercise, the polynomial f(x) = x^3 + 7x^2 - 4x - 28 is factored into x(x + 4)(x + 7). Each bracket represents a factor that gives us a zero of the function. Here, factoring unveils the zeros without laborious calculation steps. It essentially decodes the composition of the polynomial and lays out the framework of its graphical representation, grounded in the identified zeros.