Problem 1

Question

The polynomial $P(x)=3(x-5)^{3}(x-3)(x+2)$ has degree ________. It has zeros $5,3,$ and _________. The zero 5 has multiplicity ________. and the zero 3 has multiplicity_________.

Step-by-Step Solution

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Answer

The polynomial has degree 5, zeros at 5, 3, and -2. Zero 5 has multiplicity 3 and zero 3 has multiplicity 1.

1Step 1: Identify the Factors

The polynomial is given as $P(x)=3(x-5)^{3}(x-3)(x+2)$. The factors of the polynomial are $(x-5)^3$, $(x-3)$, and $(x+2)$.

2Step 2: Determine the Degree of the Polynomial

To find the degree of the polynomial, count the total number of times $x$ is a factor. $(x-5)^3$ contributes 3 to the degree, $(x-3)$ contributes 1, and $(x+2)$ contributes 1. Adding these gives the degree: $3 + 1 + 1 = 5$. The degree of the polynomial is 5.

3Step 3: Identify the Zeros

The zeros of the polynomial are the values of $x$ that make each factor equal to zero. Solving $x-5=0$ gives $x=5$; solving $x-3=0$ gives $x=3$; solving $x+2=0$ gives $x=-2$. Thus, the zeros are 5, 3, and -2.

4Step 4: Determine the Multiplicities of the Zeros

The multiplicity of a zero is determined by the exponent of its corresponding factor. The zero 5 comes from $(x-5)^3$, so it has multiplicity 3. The zero 3 comes from $(x-3)$, which has no exponent explicitly written, indicating a multiplicity of 1.

Key Concepts

Polynomial FactorsMultiplicity of ZerosDegree of Polynomial

Polynomial Factors

Polynomials are mathematical expressions consisting of variables raised to whole number exponents and constants. One of the core aspects of handling polynomials is identifying their factors. Factors are simpler expressions or numbers that can be multiplied together to obtain the polynomial. In our polynomial example, given by $ P(x) = 3(x-5)^{3}(x-3)(x+2) $, the factors are $(x-5)^3$, $(x-3)$, and $(x+2)$. Each of these terms contributes to the form and properties of the polynomial.

When identifying polynomial factors, observe the following:

Each factor is `(x - c)`, where $c$ is a constant for the linear term $x$.
Factors determine the zeros of a polynomial, as setting each factor to zero finds the value making the polynomial equal to zero.
The constant term 3 in the polynomial is a scalar multiplier and does not affect the zeros or their multiplicities.

Identifying factors is crucial as it helps in solving polynomial equations and understanding the polynomial's graph.

Multiplicity of Zeros

The concept of zero multiplicity pertains to how 'repeated' or 'intense' a zero is in a polynomial. If a polynomial has a factor $(x-c)^n$, then $x = c$ is a zero of the polynomial with multiplicity $n$. This multiplicity indicates how many times a particular zero occurs.

In the given polynomial $P(x)=3(x-5)^{3}(x-3)(x+2)$:

The zero 5 has a multiplicity of 3 because it arises from the factor $(x-5)^3$. This means at $x = 5$, the graph of the polynomial not only touches but bounces off the x-axis.
The zero 3, from the factor $(x-3)$, has a multiplicity of 1, indicating that the graph will cross the x-axis at this point.

Lower multiplicities, like 1 or 2, often mean the graph of the polynomial will sharply intersect the x-axis. Higher multiplicities imply a more gentle approach and departure, often leading to local minima or maxima at these points.

Degree of Polynomial

The degree of a polynomial is a simple yet crucial characteristic that tells us the highest power of the variable $x$ in the polynomial when expressed as a sum of terms. It gives insight into the polynomial's fundamental shape and behavior.

To find the degree of $P(x) = 3(x-5)^{3}(x-3)(x+2)$:

Each factor $(x-c)^n$ contributes $n$ to the degree. So, $(x-5)^3$ contributes 3, $(x-3)$ contributes 1, and $(x+2)$ contributes 1.
Adding these contributions gives the total degree: $3 + 1 + 1 = 5$.

The summed degree of 5 indicates that the polynomial will be most prominently influenced by terms involving $x^5$. This degree affects the number of zeros, the number of valleys and peaks in its graph, and the end behavior of the polynomial, with possible directions changing up to four times as dictated by the degree.

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