Problem 33 Find the zeros of each function.... [FREE SOLUTION]

Chapter 6: Problem 33

Find the zeros of each function. State the multiplicity of multiple zeros. $$ y=(x-4)^{2} $$

Short Answer

Expert verified

The zero of the function $y = (x-4)^{2}$ is $x = 4$ with a multiplicity of 2.

Step by step solution

Identifying the Zero of the Function

To find the zeros of the function, set the function equal to zero. In this case, the function is $y = (x-4)^{2}$. To find the zeros, solve for $x$ in the equation $0 = (x-4)^{2}$.

Solving the Equation

Since $0 = (x-4)^{2}$, take the square root of both sides to isolate $x$. The square root of zero is zero, so the equation becomes $x - 4 = 0$. Solving for $x$ gives you $x = 4$.

Determining the Multiplicity

The zero $x = 4$ is derived from the factor $x - 4$ which is squared in the original function. This indicates that the zero $x = 4$ has a multiplicity of 2, which means it is a double root.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Zeros of Polynomial Functions

Understanding the concept of zeros of polynomial functions is fundamental in algebra. Simply put, zeros are the values of the variable, usually denoted by 'x', for which the polynomial function equals zero. Think of them as the points where the graph of the function touches or crosses the x-axis. For the function provided in the exercise, $ y = (x-4)^{2} $ , we find its zero by setting it equal to zero and solving for x, which gives us $ x = 4 $. The zero at $ x = 4 $ means that the graph touches the x-axis at this point.

Multiplicity of Zeros

Zeros can appear more than once in a function, and this is where multiplicity comes into play. Multiplicity refers to the number of times a particular zero occurs. If a factor, like $x - a$, is raised to a power, that power is the multiplicity of the root $x = a$. For instance, in our example, the zero $x = 4$ is derived from the squared term $ (x-4)^{2} $ and thus has a multiplicity of 2. This tells us not only that the graph touches the x-axis at $x = 4$, but also how it behaves near this point: it simply touches the x-axis and turns back, as opposed to crossing it, which would occur if the multiplicity were odd.

Polynomial Roots

The term roots is often used interchangeably with zeros. However, the roots of a polynomial are usually discussed when we think of the function in its factored form. The roots are the solutions to the equation when it is set equal to zero. They represent the x-values where the polynomial equals zero. In our function $ y = (x-4)^{2} $, the root is $x = 4$, and this is the only root since it's the only x-value that makes the polynomial zero. The root corresponds to the x-intercept of the function, which is an essential concept in graphing polynomials.

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91影视

Find the zeros of each function. State the multiplicity of multiple zeros. $$ y=(x-4)^{2} $$

Short Answer

Step by step solution

Identifying the Zero of the Function

Solving the Equation

Determining the Multiplicity

Key Concepts

Zeros of Polynomial Functions

Multiplicity of Zeros

Polynomial Roots

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