MTH120 College Algebra Chapter 5.5

 5.5 Zeros of Polynomial Functions:

The zeros of a polynomial function, also known as its roots or x-intercepts, are the values of $�$ for which the polynomial equals zero. In other words, they are the values that make the polynomial function equal to zero. The zeros of a polynomial function can be found by solving the polynomial equation $�(�)=0$, where $�(�)$ is the polynomial.

For example, if you have a polynomial function $�(�)={�}^3-6{�}^2+9�$, finding its zeros involves solving the equation ${�}^3-6{�}^2+9�=0$ to determine the values of $�$ that make the polynomial equal to zero.

Here's how you can find the zeros of a polynomial function:

1. Write down the polynomial function $�(�)$.

2. Set $�(�)=0$ to create the polynomial equation $�(�)=0$.

3. Solve the polynomial equation for $�$ to find the values that satisfy it.

4. The solutions to the equation ($�$ values) are the zeros of the polynomial function.

It's important to note that the number of zeros a polynomial has is equal to its degree. For example, a cubic polynomial (degree 3) will have three zeros, either real or complex.

Additionally, the Fundamental Theorem of Algebra states that every polynomial of degree $�$ (where $�$ is a positive integer) has exactly $�$ complex zeros, counting multiplicity. This means that even if some zeros are complex, they will occur in conjugate pairs.

For example, if you have a quadratic polynomial, $�{�}^2+��+�$, its zeros can be found using the quadratic formula:

$�=\frac{-�\pm \sqrt{{�}^2-4��}}{2�}$

The quadratic formula will yield two values for $�$, which are the zeros of the quadratic polynomial. If the discriminant (${�}^2-4��$) is negative, the zeros will be complex conjugates.

In summary, finding the zeros of a polynomial involves solving the polynomial equation $�(�)=0$, and the number of zeros is equal to the degree of the polynomial. Complex zeros, if they exist, will occur in conjugate pairs.

The Remainder Theorem is a fundamental concept in algebra that provides a way to evaluate a polynomial function at a specific value without having to fully simplify the polynomial. It states that when you divide a polynomial $�(�)$ by the binomial $(�-�)$, the remainder of this division will be equal to $�(�)$, where $�$ is a constant.

In other words, if you want to find the value of a polynomial $�(�)$ at a particular $�=�$, you can evaluate it by finding the remainder when $�(�)$ is divided by $(�-�)$.

Here's how to use the Remainder Theorem to evaluate a polynomial at a specific value:

1. Write down the polynomial $�(�)$ that you want to evaluate.

2. Set up the division: Divide $�(�)$ by $(�-�)$, where $�$ is the value at which you want to evaluate the polynomial.

3. Perform synthetic division or long division to divide $�(�)$ by $(�-�)$.

4. The remainder you obtain from the division is $�(�)$, which is the value of the polynomial at $�=�$.

Here's an example:

Problem: Evaluate the polynomial $�(�)=2{�}^3-5{�}^2+3�-1$ at $�=3$.

Step 1: Write down the polynomial $�(�)$: $�(�)=2{�}^3-5{�}^2+3�-1$

Step 2: Set up the division: Divide $�(�)$ by $(�-3)$ since we want to evaluate it at $�=3$.

Step 3: Perform synthetic division:

            2   -5    3   -1
        ----------------------
(x - 3) |  2    -5    3   -1
            6     3    18
        ----------------------
              2     1   21

Step 4: The remainder from the division is $�(3)=2$, which is the value of the polynomial $�(�)$ at $�=3$.

So, $�(3)=2$.

This demonstrates how to use the Remainder Theorem to evaluate a polynomial at a specific value of $�$. In this case, when $�=3$, $�(�)=2$.

The Remainder Theorem is a fundamental concept in algebra that deals with the remainder when you divide a polynomial by a binomial of the form $(�-�)$, where $�$ is a constant. The theorem states that if you divide a polynomial $�(�)$ by $(�-�)$, the remainder will be equal to $�(�)$. In other words, when you substitute $�=�$ into the polynomial, the result is the remainder of the division.

Mathematically, the Remainder Theorem can be expressed as follows:

If $�(�)$ is a polynomial and you divide it by $(�-�)$, the remainder $�$ is given by:

$�=�(�)$

Here's how to use the Remainder Theorem:

1. Write down the polynomial $�(�)$ that you want to divide.

2. Choose a value for $�$.

3. Substitute $�=�$ into $�(�)$ to find $�(�)$.

4. The result $�(�)$ is the remainder when $�(�)$ is divided by $(�-�)$.

The Remainder Theorem is often used in conjunction with the Factor Theorem to find factors and zeros (roots) of polynomial equations. By evaluating a polynomial at a specific value ($�$), you can determine whether $(�-�)$ is a factor of the polynomial. If $�(�)=0$, then $(�-�)$ is a factor, and $�$ is a zero of the polynomial.

For example, if you have a polynomial $�(�)$ and you want to check whether $(�-2)$ is a factor, you can use the Remainder Theorem by evaluating $�(2)$. If $�(2)=0$, then $(�-2)$ is a factor, and $�=2$ is a zero of the polynomial.

The Remainder Theorem is a powerful tool in algebra for understanding the properties of polynomials and finding their factors and zeros. It simplifies the process of polynomial evaluation and plays a crucial role in polynomial factorization.

The Factor Theorem is a fundamental concept in algebra that helps you determine whether a given binomial $(�-�)$ is a factor of a polynomial and, if it is, find the corresponding zeros (roots) of the polynomial equation. The Factor Theorem is closely related to the Remainder Theorem and is used to factor polynomials.

Here's how to use the Factor Theorem to solve a polynomial equation:

Step 1: Write Down the Polynomial Equation

Start with the polynomial equation you want to solve. For example, let's say you have the equation:

$�(�)={�}^3-6{�}^2+11�-6=0$

Step 2: Choose a Value for $�$

Select a value for $�$ such that $(�-�)$ is a potential factor of the polynomial. In the Factor Theorem, $�$ is the potential zero you want to test. Typically, you would choose small integers or rational numbers to test first.

Step 3: Apply the Factor Theorem

Evaluate the polynomial $�(�)$ at the chosen value $�$. If $�(�)=0$, it means that $(�-�)$ is a factor of the polynomial, and $�$ is a zero (root) of the polynomial. In other words, if $�(�)=0$, the binomial $(�-�)$ divides evenly into the polynomial.

For example, if you choose $�=2$ and find that $�(2)=0$, it indicates that $(�-2)$ is a factor of the polynomial $�(�)$, and $�=2$ is a zero.

Step 4: Repeat as Necessary

If the chosen value $�$ does not result in $�(�)=0$, try another value for $�$ and repeat the process. Continue until you find a value of $�$ that makes $�(�)=0$.

Step 5: Factor the Polynomial

Once you've found a value of $�$ that makes $�(�)=0$, you can use long division or synthetic division to divide the polynomial by $(�-�)$ to find the other factors of the polynomial.

For example, if $�=2$ is a zero of $�(�)$, then $(�-2)$ is a factor. Divide $�(�)$ by $(�-2)$ to find the other factors and zeros.

This process allows you to factor the polynomial completely and find all of its zeros, solving the polynomial equation.

The Factor Theorem is a powerful tool for finding zeros of polynomial equations and factoring polynomials, especially when dealing with higher-degree polynomials. It simplifies the process of factorization and helps you determine whether a given value is a zero of the polynomial.

The Rational Zero Theorem, also known as the Rational Root Theorem, is a valuable tool for finding the rational zeros (roots) of a polynomial equation. It provides a systematic way to identify potential rational zeros from the coefficients of the polynomial. Here's how to use the Rational Zero Theorem:

Step 1: Write Down the Polynomial Equation

Start with the polynomial equation you want to solve. For example, consider the equation:

$�(�)=2{�}^3-3{�}^2-11�+6=0$

Step 2: Identify the Coefficients of the Polynomial

Identify the coefficients of the polynomial $�(�)$. In this example, the coefficients are:

• Leading coefficient (the coefficient of the highest-degree term): 2
• Coefficient of the second-highest-degree term: -3
• Coefficient of the second-degree term: -11
• Constant term: 6

Step 3: Apply the Rational Zero Theorem

The Rational Zero Theorem states that if a rational zero $�\mathrm{/}�$ exists for the polynomial equation $�(�)=0$, where $�$ is a factor of the constant term (6 in this case) and $�$ is a factor of the leading coefficient (2 in this case), then $�\mathrm{/}�$ is a potential rational zero.

In other words, potential rational zeros are in the form $\pm \frac{�}{�}$, where $�$ is a factor of 6 (1, 2, 3, or 6), and $�$ is a factor of 2 (1 or 2).

Step 4: List the Potential Rational Zeros

List all the potential rational zeros based on the factors of the constant term and the leading coefficient. In this example, the potential rational zeros are:

$\pm 1,\pm 2,\pm 3,\pm 6$

Step 5: Test the Potential Zeros

For each potential rational zero, evaluate $�(�)$ by substituting it into the polynomial equation. If $�$ equals zero for a specific potential zero, then that value is a rational zero of the polynomial.

• Test $�=1$: $�(1)=2(1{)}^3-3(1{)}^2-11(1)+6=2-3-11+6=-6\mathrm{\ne }0$

• Test $�=-1$: $�(-1)=2(-1{)}^3-3(-1{)}^2-11(-1)+6=-2-3+11+6=12\mathrm{\ne }0$

Continue testing each potential zero until you find one that makes $�(�)=0$.

Step 6: Find the Rational Zeros

After testing all the potential rational zeros, you'll find the ones that make (P(x) = 0. These are the rational zeros of the polynomial equation.

In this example, the Rational Zero Theorem might reveal that $�=2$ is a rational zero, and further polynomial division or synthetic division can be used to find the remaining zeros of the polynomial equation.

Using the Rational Zero Theorem helps narrow down the search for rational zeros, making it a useful tool for solving polynomial equations, especially when dealing with higher-degree polynomials.

Finding the zeros of a polynomial function is a crucial task in algebra. The zeros, also known as roots or x-intercepts, are the values of $�$ for which the polynomial equals zero. To find the zeros of a polynomial function, you can follow these steps:

Step 1: Write Down the Polynomial Function

Start with the polynomial function you want to analyze. For example, consider the polynomial function:

$�(�)=2{�}^3-3{�}^2-11�+6$

Step 2: Set Up the Polynomial Equation

Set the polynomial function equal to zero:

$2{�}^3-3{�}^2-11�+6=0$

Step 3: Use Factoring

If possible, try to factor the polynomial equation. Factoring can help you identify the zeros directly. In some cases, you may be able to factor out common factors or use special factoring techniques like the difference of squares, sum/difference of cubes, or grouping.

In our example, the polynomial doesn't factor easily.

Step 4: Apply the Rational Zero Theorem

Use the Rational Zero Theorem to identify potential rational zeros. According to the theorem, rational zeros (if they exist) are in the form of $\pm \frac{�}{�}$, where $�$ is a factor of the constant term (6) and $�$ is a factor of the leading coefficient (2).

Potential rational zeros for our example include:

$\pm 1,\pm 2,\pm 3,\pm 6$

Step 5: Test Potential Zeros

Substitute each potential zero into the polynomial equation to see if it equals zero. This process may involve polynomial long division or synthetic division to simplify the calculations.

• Test $�=1$: $�(1)=2(1{)}^3-3(1{)}^2-11(1)+6=2-3-11+6=-6\mathrm{\ne }0$

• Test $�=-1$: $�(-1)=2(-1{)}^3-3(-1{)}^2-11(-1)+6=-2-3+11+6=12\mathrm{\ne }0$

Continue testing all the potential zeros until you find one that makes $�(�)=0$.

Step 6: Solve for Zeros

Once you find a value of $�$ that makes $�(�)=0$, it represents a zero (root) of the polynomial function. In our example, let's assume that you found $�=2$ as one of the zeros.

Step 7: Divide the Polynomial

After identifying one zero, you can perform polynomial division or synthetic division to factor the polynomial further and find the remaining zeros. In this case, you'd divide $�(�)$ by $(�-2)$ to find the other zeros.

By following these steps, you can systematically find the zeros of a polynomial function, whether they are rational or irrational. Polynomial division techniques help factor the polynomial and find all its roots, providing a complete solution to the equation.

The Fundamental Theorem of Algebra is a fundamental result in mathematics that states that every non-constant polynomial equation with complex coefficients has at least one complex root. In other words, it guarantees that polynomial equations of the form $�(�)=0$, where $�(�)$ is a polynomial with complex coefficients, always have at least one solution in the complex numbers.

Here's how you can use the Fundamental Theorem of Algebra:

Step 1: Formulate the Polynomial Equation

Start with the polynomial equation you want to analyze. This equation should be of the form $�(�)=0$, where $�(�)$ is a non-constant polynomial with complex coefficients.

For example, consider the polynomial equation:

$�(�)={�}^2-5�+6=0$

Step 2: Recognize the Complex Coefficients

Make sure that the coefficients of the polynomial are complex numbers or real numbers. Complex numbers include both real and imaginary parts. In the example above, the coefficients (1, -5, 6) are real numbers, and complex numbers can be seen as real numbers with an imaginary part of zero.

Step 3: Apply the Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra guarantees that there is at least one complex number $�$ that satisfies the equation $�(�)=0$.

In the example, you can use this theorem to conclude that the equation ${�}^2-5�+6=0$ has at least one complex solution.

Step 4: Solve for the Complex Roots

To find the complex roots of the polynomial equation, you can use various methods, such as factoring, the quadratic formula (for quadratic polynomials), synthetic division, or numerical methods. In the case of ${�}^2-5�+6=0$, you can factor it as $(�-2)(�-3)=0$ and find that $�=2$ and $�=3$ are the complex roots.

It's important to note that the Fundamental Theorem of Algebra guarantees at least one complex root but doesn't specify how many complex roots there are in total. Higher-degree polynomials may have multiple complex roots, some of which may be repeated. Complex roots come in conjugate pairs, so if $�+��$ is a complex root, then $�-��$ is also a root.

In summary, the Fundamental Theorem of Algebra ensures that any non-constant polynomial equation with complex coefficients has at least one complex root. To find the complex roots, you may need to use various algebraic techniques depending on the degree of the polynomial.

The Linear Factorization Theorem states that every polynomial of degree $�$ with complex coefficients can be factored into a product of linear factors. This theorem provides a way to find a polynomial when you are given its zeros (roots).

Here's how to use the Linear Factorization Theorem to find a polynomial with given zeros:

Step 1: Start with the Zeros

Begin by listing the zeros (roots) of the polynomial. These are the values of $�$ for which the polynomial equals zero. Let's assume you have the following zeros: $�=�$, $�=�$, $�=�$, and so on.

For example, consider the zeros $�=2$, $�=-3$, and $�=5$.

Step 2: Set Up the Linear Factors

Using the zeros, set up linear factors for the polynomial. Each linear factor has the form $(�-\text{zero})$. So, in our example, the linear factors would be:

$(�-2)$, $(�+3)$, and $(�-5)$

Step 3: Multiply the Linear Factors

Multiply all the linear factors together to obtain the polynomial. This will be the polynomial that has the given zeros. Here's how it looks in our example:

$(�-2)(�+3)(�-5)$

Step 4: Expand and Simplify

Expand and simplify the polynomial if needed. In this case, you can expand the expression and simplify the result:

$(�-2)(�+3)(�-5)=({�}^2+3�-2�-6)(�-5)=({�}^2+�-6)(�-5)$

Step 5: Verify

Verify that the polynomial you found indeed has the specified zeros. You can do this by substituting the given zeros into the polynomial and confirming that the result is zero.

For example, substitute $�=2$, $�=-3$, and $�=5$ into the polynomial $({�}^2+�-6)(�-5)$ and check that it equals zero for each of these values.

Using the Linear Factorization Theorem allows you to construct a polynomial with given zeros by expressing it as a product of linear factors. This is a valuable tool for solving polynomial problems and understanding the relationship between zeros and factors of a polynomial.

The Complex Conjugate Theorem, often referred to as the Complex Conjugate Root Theorem, is a fundamental theorem in algebra that provides information about the roots (zeros) of a polynomial equation with real coefficients.

The Complex Conjugate Theorem states the following:

If a polynomial equation with real coefficients has a complex root $�+��$, where $�$ and $�$ are real numbers and $�$ is the imaginary unit (${�}^2=-1$), then its complex conjugate $�-��$ is also a root of the polynomial.

In other words, complex roots of a polynomial equation with real coefficients always occur in conjugate pairs.

Here's how to use the Complex Conjugate Theorem:

Step 1: Identify Complex Roots

If you have a polynomial equation with real coefficients and you find a complex root $�+��$ where $�$ and $�$ are real numbers and $�$ is the imaginary unit, you can conclude that $�-��$ is also a root.

Step 2: Use Conjugate Pairs

When dealing with polynomial equations, you can work with either of the complex conjugate roots to factor the equation or find the complete set of roots. If you find one complex root, you automatically know its conjugate is also a root.

For example, if you have a polynomial equation with real coefficients and you find the complex root $3+2�$, you can immediately conclude that $3-2�$ is also a root of the equation.

The Complex Conjugate Theorem simplifies the process of finding all the roots of a polynomial equation with real coefficients. It guarantees that complex roots come in pairs, which helps when dealing with complex numbers in algebraic computations.

Descartes' Rule of Signs is a useful tool for analyzing the possible number of positive and negative real roots (zeros) of a polynomial equation with real coefficients. It helps you determine the maximum number of positive and negative roots without necessarily finding the exact values of those roots.

Here's how to use Descartes' Rule of Signs:

Step 1: Write Down the Polynomial Equation

Start with the polynomial equation you want to analyze. Ensure that it is in standard form, with the terms arranged in descending order of degree.

For example, consider the polynomial equation:

$�(�)=2{�}^4-3{�}^3+5{�}^2-7�+1=0$

Step 2: Count the Sign Changes in $�(�)$

Count the number of sign changes in the coefficients of the polynomial as you move from left to right. Write down the sequence of signs you observe.

In our example, the sign changes are as follows:

From $2{�}^4$ to $-3{�}^3$, there is a change in sign (positive to negative). From $-3{�}^3$ to $5{�}^2$, there is a change in sign (negative to positive). From $5{�}^2$ to $-7�$, there is a change in sign (positive to negative). From $-7�$ to $1$, there is a change in sign (negative to positive). The number of sign changes is 4.

Step 3: Determine the Possible Number of Positive Roots

The number of sign changes in $�(�)$ represents the possible number of positive real roots or the maximum number of positive real roots. In our example, there are 4 sign changes, so there can be either 4 positive real roots or fewer than 4 positive real roots.

Keep in mind that Descartes' Rule of Signs doesn't tell you the exact number of positive real roots but gives you an upper limit.

Step 4: Analyze the Alternating Sign

Next, consider the alternation in sign when you look at the coefficients with their absolute values.

In our example, we have:

The absolute value of the coefficient for ${�}^4$ is $2$. The absolute value of the coefficient for ${�}^3$ is $3$. The absolute value of the coefficient for ${�}^2$ is $5$. The absolute value of the coefficient for $�$ is $7$. The absolute value of the constant term is $1$. Now, count the sign changes in this sequence:

There is no sign change for $2$ to $3$ (both positive). There is a sign change for $3$ to $5$ (positive to positive). There is no sign change for $5$ to $7$ (positive to positive). There is no sign change for $7$ to $1$ (positive to positive). The number of sign changes in this sequence is $1$.

Step 5: Determine the Possible Number of Negative Roots

The number of sign changes in the sequence of absolute values represents the possible number of negative real roots or the maximum number of negative real roots. In our example, there is $1$ sign change, so there can be either $1$ negative real root or fewer than $1$ negative real root.

Step 6: Interpret the Results

Based on Descartes' Rule of Signs for our example polynomial $�(�)$, there can be:

A maximum of $4$ positive real roots. A maximum of $1$ negative real root.

Please note that these results provide upper bounds on the possible number of real roots but do not specify the exact values or locations of those roots. The polynomial may have fewer real roots than indicated by the rule, and it may also have complex roots. Further analysis or techniques, such as the Rational Zero Theorem or synthetic division, may be needed to determine the exact roots.

Descartes' Rule of Signs can be applied to real-world scenarios involving polynomial equations to analyze the possible number of positive and negative real roots. Let's consider a real-world example involving a financial situation.

Real-World Example: Investment Growth

Suppose you are an investor looking to invest in a project that promises to generate positive returns over time. You have a polynomial equation that models the investment's growth over time, taking into account certain factors. The polynomial equation is as follows:

$�(�)=3{�}^3-2{�}^2-7�+1=0$

Here, $�(�)$ represents the value of your investment at time $�$ years from now. You want to use Descartes' Rule of Signs to analyze the possible number of positive and negative real roots of this polynomial equation.

Step 1: Write Down the Polynomial Equation

The given polynomial equation represents the value of your investment over time.

Step 2: Count the Sign Changes in $�(�)$

Now, count the sign changes in the coefficients of the polynomial as you move from left to right:

• From $3{�}^3$ to $-2{�}^2$, there is a change in sign (positive to negative).
• From $-2{�}^2$ to $-7�$, there is no change in sign (both negative).
• From $-7�$ to $1$, there is a change in sign (negative to positive).

The number of sign changes is $2$.

Step 3: Determine the Possible Number of Positive Roots

The number of sign changes in $�(�)$ represents the possible number of positive real roots or the maximum number of positive real roots. In this case, there can be either $2$ positive real roots or fewer than $2$ positive real roots.

Step 4: Analyze the Alternating Sign

Consider the alternation in sign when you look at the coefficients with their absolute values:

• The absolute value of the coefficient for ${�}^3$ is $3$.
• The absolute value of the coefficient for ${�}^2$ is $2$.
• The absolute value of the coefficient for $�$ is $7$.
• The absolute value of the constant term is $1$.

Now, count the sign changes in this sequence:

• There is a sign change for $3$ to $2$ (positive to positive).
• There is a sign change for $2$ to $7$ (positive to positive).
• There is no sign change for $7$ to $1$ (both positive).

The number of sign changes in this sequence is $2$.

Step 5: Determine the Possible Number of Negative Roots

The number of sign changes in the sequence of absolute values represents the possible number of negative real roots or the maximum number of negative real roots. In this case, there can be either $2$ negative real roots or fewer than $2$ negative real roots.

Step 6: Interpret the Results

Based on Descartes' Rule of Signs for the investment growth polynomial equation:

• There can be a maximum of $2$ positive real roots, representing potential points in time when your investment shows positive growth.
• There can be a maximum of $2$ negative real roots, representing potential points in time when your investment experiences temporary losses or downturns.

These results provide insights into the possible behavior of your investment over time. However, to determine the exact times when these changes occur and the corresponding investment values, further analysis or numerical methods may be necessary.

In the real world, Descartes' Rule of Signs can be applied to various scenarios involving polynomial equations, such as economic modeling, population growth, and physical phenomena, to analyze the behavior and predict potential outcomes.