Problem 27 - 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. $$f(x)=4(x-3)(x+6)^{3}$$

Step-by-Step Solution

Answer

The zero of the function are \(x = 3\) and \(x = -6\) with multiplicities of 1 and 3, respectively. The graph crosses the x-axis at both zeros.

Step 1: Set function to zero

To find the zeros of a function, set the function equal to zero: \(4(x-3)(x+6)^3 = 0\).

Step 2: Solve for x

Solving for x, two possible values are found. These are \(x = 3\) when \((x-3) = 0\), and \(x = -6\) when \((x+6) = 0\).

Step 3: State multiplicities and graph behavior

The function \(x-3\) has a multiplicity of 1 since the exponent of the factor is 1. This means the graph crosses the x-axis at \(x = 3\). On the other hand, the function \((x+6)^3\) has a multiplicity of 3, an odd multiplicity. This suggests that the graph will also crosses the x-axis at \(x = -6\).

Key Concepts

Multiplicity of Zeros

When discussing polynomial functions, the term "multiplicity of zeros" refers to the number of times a particular zero appears. In the expression \((x - a)^m\), the zero \(x = a\) has a multiplicity \(m\). For example, in the polynomial \(f(x) = (x-3)(x+6)^3\), we have two zeros: \(x = 3\) and \(x = -6\). The zero \(x = 3\) appears once, so its multiplicity is 1. The zero \(x = -6\) appears three times, thus it has a multiplicity of 3.
Understanding the multiplicity is important because it tells us how the graph behaves at those zeros. Smaller multiplicities indicate a quicker crossing or touching of the x-axis, while larger ones indicate a more gradual approach or departure.

Crossing the X-Axis

When examining polynomial functions, the graph crosses the x-axis at zeros where the multiplicity is odd. This means that the function changes sign as it passes through these points. In our example \(f(x) = (x-3)(x+6)^3\), the zero \(x = 3\) has a multiplicity of 1, which is odd.
This tells us that the graph crosses the x-axis at this point. Similarly, when considering \(x = -6\), where the multiplicity is 3 (also odd), the graph once again crosses the x-axis.
A crossing indicates that the graph goes from above to below the x-axis or vice versa. This transition is evident in polynomial functions with odd multiplicities.

Touching the X-Axis

Touching the x-axis without crossing occurs when the zero has an even multiplicity. This behavior is often described as "bouncing" or "turning around" at the x-axis. However, in our example, both zeros have odd multiplicities (1 and 3, respectively). So there is no instance of the graph merely touching the x-axis.
If we imagine a scenario where there was a factor such as \((x+2)^2\), the zero \(x = -2\) would have a multiplicity of 2. In that case, the graph would touch the x-axis at \(x = -2\) and turn back in the opposite direction, rather than crossing it.
This action at even multiplicities results in the polynomial function momentarily approaching the x-axis but not transitioning through it.

Polynomial Functions

Polynomial functions are expressions consisting of variables and coefficients, often linked by operations such as addition, subtraction, and multiplication. They are generally expressed in the form \(P(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0\), where the exponents are non-negative integers.
The degree of a polynomial function is the highest power of the variable found in the expression. It indicates the maximum number of zeros the polynomial can have. In the given function \(f(x) = 4(x-3)(x+6)^3\), the degree is 4 (1 from \((x-3)\) and 3 from \((x+6)^3\)). This suggests that up to 4 zeros exist, considering multiplicities.
Understanding polynomial functions and their behavior at zeros give insight into the overall shape and direction of their graphs, which is crucial in solving real-world problems involving such functions.