Problem 32 - 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. $$f(x)=x^{3}+5 x^{2}-9 x-45$$

Step-by-Step Solution

Answer

The zeros of the function \(f(x)=x^{3}+5 x^{2}-9 x-45\) are \(x=0\), \(x=-1\), and \(x=-5\), each with a multiplicity of 1. The graph of the function crosses the \(x\)-axis at each of these points.

Step 1: Setting the Function Equal to Zero

First thing to do is to set the function equal to zero and solve for \(x\). This gives: \[x^{3}+5 x^{2}-9 x-45 = 0 \]

Step 2: Factoring the Polynomial

Factor the polynomial to find the roots easier. This can be done by grouping or using other factoring methods. For this function, it can be factored as follows: \[x(x+1)(x+5) = 0\]

Step 3: Identifying the Zeros

The zeros of the polynomial function are the values of \(x\) that make the expression equal to zero. To get them, each factor is set to zero and solved for \(x\). Here, those would be: \[x=0, x=-1, and x=-5\]

Step 4: Determining the Multiplicity and Behavior at each Zero

The multiplicity of each zero is the power of the corresponding \(x\)-term in the factored polynomial. Here, each zero has a multiplicity of 1. All the roots have odd multiplicities, so the graph will cross the \(x\)-axis at all the zeros.

Key Concepts

Factoring Polynomials

Factoring polynomials is a crucial step in finding the zeros of a polynomial function. By breaking down a polynomial into simpler components, we can solve for values of \( x \) that make the equation zero, known as the zeros of the polynomial.

The process often involves factoring by grouping, using special formulas, or employing the trial and error method.

• Factoring by grouping involves rearranging terms and finding common factors in small groups. This method is useful when dealing with polynomials that can be split into sub-polynomials that share a common factor.
• Special formulas like the difference of squares, perfect square trinomials, or the sum/difference of cubes can also simplify factoring. These formulas are shortcuts based on specific patterns in the polynomial’s terms.

For the polynomial \( f(x) = x^3 + 5x^2 - 9x - 45 \), factoring by grouping results in \( x(x+1)(x+5) = 0 \).

This means the polynomial has been reduced to factors that, when multiplied together, recreate the original expression. Solving these factors gives us the zeros, which are the solutions to the equation.

Multiplicity of Zeros

The concept of multiplicity refers to the number of times a particular zero appears in the factored form of a polynomial. It indicates how many times a particular root repeats. This concept plays a significant role in understanding the behavior of polynomial graphs at these points.

When a zero's multiplicity is:

• Odd, the graph will cross the \( x \)-axis at this point.
• Even, the graph will touch the \( x \)-axis and "bounce" off, turning around at this zero.

In our example, the polynomial factors to \( x(x+1)(x+5) \). Each of \( x=0 \), \( x=-1 \), and \( x=-5 \) appears once, giving them each a multiplicity of 1.

Due to each zero having an odd multiplicity, the graph of the polynomial crosses the \( x \)-axis at all these points.

Polynomial Graph Behavior

Understanding polynomial graph behavior helps us predict how the graph will look based solely on its equation. Specifically, the behavior of a graph around its zeros is influenced by the multiplicity of those zeros.

Here's how the behavior is determined:

• If a zero has odd multiplicity, the graph crosses the \( x \)-axis. For example, a zero with multiplicity of 1 will have the graph intersect the \( x \)-axis, indicating a change in the sign of the function.
• If the zero has even multiplicity, the graph touches and turns around at the \( x \)-axis. This means the function does not change its sign.

Using \( f(x) = x^3 + 5x^2 - 9x - 45 \) as an example, and having identified the zeros at \( x=0 \), \( x=-1 \), and \( x=-5 \), we see odd multiplicities for each zero.

Thus, the graph of this polynomial will cross the \( x \)-axis at these points, confirming that each zero results from a root crossing through the axis without turning back. This understanding can be quite intuitive once you practice identifying multiplicities and predicting graph behavior accordingly.