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# Real Zeros of Polynomials

## Remainder, Factor, and Rational Zero Theorems and Descartes' Rule of Signs.

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Real Zeros of Polynomials

In the real world, problems do not always easily fit into quadratic or even cubic equations. Financial models, population models, fluid activity, etc., all often require many degrees of the input variable in order to approximate the overall behavior. While it can be challenging to model some of these more complex interactions, the effort can be well worth it. Mathematical models of stocks are used constantly as a way to "look into the future" of finance and make the kinds of educated guesses that are behind some of the largest fortunes in the world.

What benefits can you think of to modeling the behavior of large populations? Can you think of other useful applications not mentioned here?

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### Guidance

There are three theorems and a rule that we will be referring to during this lesson in order to help make the discovery of the roots of polynomial functions easier. You should review them and be prepared to refer to them often during the practice problems.

The Remainder Theorem

If a polynomial of degree is divided by , then the remainder is a constant and it is equal to the value of the polynomial when is substituted for . That is

The Factor Theorem

If is a polynomial of degree and , then is a factor of the polynomial . Further, if is a factor, then is a zero of .

The Rational Zero Theorem

Given the polynomial

and is a positive integer. If the coefficients are integers and is a rational zero in lowest terms, then is a divisor of and is a divisor of .

Descartes Rule of Signs

Given any polynomial, ,

1. Write it with the terms in descending order, i.e. from the highest degree term to the lowest degree term.
2. Count the number of sign changes of the terms in . Call the number of sign changes .
3. Then the number of positive roots of is less than or equal to .
4. Further, the possible number of positive roots is
5. To find the number of negative roots of , write in descending order as above (i.e. change the sign of all terms in with odd powers), and repeat the process above. Then the maximum number of negative roots is .

#### Example A

Use synthetic division and the remainder and factor theorems to find the quotient and the remainder if is divided by .

Solution

Hence

Notice that the remainder is 181 and it can also be obtained if we simply substituted into ,

#### Example B

Use the rational zero theorem and synthetic division to find all the possible rational zeros of the polynomial

Solution

From the rational zero theorem, is a rational zero of the polynomial . So is a divisor of 2 and is a divisor of 1. Hence, can take the following values: -1, 1, -2, 2 and can be either -1 or 1. Therefore, the possible values of are

So there are four possible zeros. Of these four, not more than three can be zeros of because is a polynomial with degree 3. To test which of the four possible candidates are zeros of , we use the synthetic division. Recall from the remainder theorem if , then is a zero of . We have

Hence, 2 is a zero of . Further, by the division algorithm,

The remaining zeros of are simply the zeros of which is easier to manipulate,

and thus the remaining zeros are -1 and 1. Thus the rational zeros of are -1, 1, and 2.

#### Example C

Graph the polynomial function .

Solution

Notice that the leading term is , where odd and . This tells us that the end behavior will take the shape of a power function with an odd exponent.

Here, as you can see, there is no straight-forward way to find the zeros of . However, with the use of the factor theorem and the synthetic division, we can find the rational roots of .

First, we use the rational zero theorem and find that the possible rational zeros are

testing all these numbers by the synthetic division,

-1 is not a root. Now let's test .

we find that 1 is a zero of and so we can re-write ,

and so

Thus 1 and are the intercepts of . The intercept is

Further, the synthetic division can be also used to form a table of values for the graph of :

We choose test points from each interval and find .

Interval Test Value Sign of Location of points on the graph
-1 -28 - below the axis
- below the axis
3 4 + above the axis

From this information, the graph of is shown in the two graphs below. Notice that the second graph is a magnification of in the vicinity of the axis.

Were you able to identify some valuable real-world uses for modeling higher-degree polynomials? Here are a few possibilities:

Identifying what times of the day people are most likely to want coffee (market research like this is key to running your own business)
Predicting population growth in a particular neighborhood or area of town (useful for identifying a good location to start a small business)
Identifying which stocks are likely to rise or fall based on weather or season
Forecasting the weather
Calculating the right head start to give a slower car to make a drag race exciting

There are many, many more.

### Guided Practice

Questions:

1) Show that is a factor of .

Find the quotient and express in factored form.

2) Use the 'rational zero' theorem and synthetic division to find all the possible rational zeros of the polynomial

3) Use Descartes Rule of Signs to identify the possible number of positive and negative roots of

.

4) Find the root(s) of

5) Find the root(s) of

Solutions

1) By the factor theorem, if , then is a factor of the polynomial. In other words, if the synthetic division produces a remainder equal to zero, then is a factor of the polynomial. :Using the synthetic division with ,

Hence, , and the quotient is
so that can be written as

2) Assume is a rational zero of . By the rational zero theorem, is a divisor of 6 and is a divisor of 1. Thus and can assume the following respective values

and
Therefore, the possible rational zeros will be
Notice that with these choices for and there could be rational zeros. But, eight of them are duplicates. For example . The next step is to test all these values by the synthetic division (we'll let you do this on your own for practice) and we finally find that
are zeros of . That is

3) First, re-write in descending order

The number of sign changes of is 2, so the number of positive roots is either 2 or 0.
For the negative roots, write
The number of sign changes of is 2, so the maximum number of negative roots is 2.
The graph of below shows that there is one negative root and two positive roots.

4) Solve by factoring and applying the zero product rule:

or
or
(each zero has a multiplicity of one)

5) This one is easy:

Since there are no real roots of even powers, this function has zero real solutions.

### Practice

Problems 1 - 3: Use a) long division and b) synthetic division to perform the divisions.

Express each result in the form:
1. by
2. by
3. by
4. Use synthetic division to find and so that if and
5. If , use synthetic division to determine the following: a) b) c) d) e) What are the factors of ?
6. Find so that is a factor of
7. Use synthetic division to determine all the zeros of the polynomials: a) b)
8. Graph the polynomial function by using synthetic division to find the intercepts and locate the intercepts.
9. Graph the polynomial function by using synthetic division to find the intercepts and locate the intercepts.
10. Write a 3rd degree equation of a polynomial function with the zeroes: 0, 2, and -5.
11. Write a 7th degree equation of a polynomial function with the zeroes: 0 (multiplicity 2), 2 (multiplicity 3), and -5 (multiplicity 2)
12. Write a quadratic equation which has 4 (multiplicity 2) as the zero and opens downward.
13. Write a 3rd degree polynomial function with the zeroes: -2, 2, and 6, passing through the point (3, 4)
14. Let and find the solutions: a) b)
15. Graph and find the solution set of the inequality .
16. Use the graph of to find the solution set of the inequality .

### Vocabulary Language: English

Descartes' Rule of Signs

Descartes' Rule of Signs

Descartes' rule of signs is a technique for determining the number of positive and negative real roots of a polynomial.
factor theorem

factor theorem

The factor theorem states that if $f(x)$ is a polynomial of degree $n>0$ and $f(c)=0$, then $x-c$ is a factor of the polynomial $f(x)$.
factorization theorem

factorization theorem

The factorization theorem states that If $f(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots+a_{1}x+a_{0}$, where $a_{n} \ne 0$, and $n$ is a positive integer, then $f(x)=a_{n}(x-c_{1})(x-c_{2})\cdots(x-c_{0})$ where the numbers $c_{i}$ are complex numbers.
Multiplicity

Multiplicity

The multiplicity of a term describes the number of times the given term acts as a zero of the given function.
Polynomial

Polynomial

A polynomial is an expression with at least one algebraic term, but which does not indicate division by a variable or contain variables with fractional exponents.
Rational Zero Theorem

Rational Zero Theorem

The rational zero theorem states that for a polynomial, $f(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$, where $a_n, a_{n-1}, \cdots a_0$ are integers, the rational roots can be determined from the factors of $a_n$ and $a_0$. More specifically, if $p$ is a factor of $a_0$ and $q$ is a factor of $a_n$, then all the rational factors will have the form $\pm \frac{p}{q}$.
Remainder Theorem

Remainder Theorem

The remainder theorem states that if $f(k) = r$, then $r$ is the remainder when dividing $f(x)$ by $(x - k)$.
Roots

Roots

The roots of a function are the values of x that make y equal to zero.
Synthetic Division

Synthetic Division

Synthetic division is a shorthand version of polynomial long division where only the coefficients of the polynomial are used.
Zeroes

Zeroes

The zeroes of a function $f(x)$ are the values of $x$ that cause $f(x)$ to be equal to zero.
Zeros

Zeros

The zeros of a function $f(x)$ are the values of $x$ that cause $f(x)$ to be equal to zero.