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Authors
W11: Steve Suzio, Laura Todd, Robert LaMothe, Grace Pushman
F11: Emily Wyble, Ryan Warner, Jessica Brown


History of Polynomials:

The earliest known people to solve polynomial equations were the Ancient Egyptians and Babylonians, although they didn't see what they were solving as equations, that terminology came later. They were able to solve linear (ax=b), quadratic (ax^2+bx=c), and indeterminate equations (x^2+y^2=z^2), and the way we solve them today is still quite similar to their methods.

Later, Alexandrian mathematicians, the Hero of Alexandria and Diophantus, took the ideas that the Egyptians and Babylonians had come up with and expanded upon them. Their knowledge became a staple of Islamic world, where it became known as "the science of restoration and balancing." The Arabic word for restoration, al-jabru, became the root for the word algebra. In the 9th century, the Arab mathematician al-Kwharizmi wrote one of the first books on Arabic algebra, and it provided examples and proofs of what we now know to be basic algebraic theory. By the end of the 9th century, another Arab mathematician, Abu Kamil, had expanded even further on al-Kwharizmi's theories and was able to prove the basic laws and identities of algebra and solve more complicated problems.

In about 300 BC
Euclid
developed a geometrical approach which, although later mathematicians used it to solve quadratic equations, amounted to finding a length which in our notation was the root of a quadratic equation. Euclid had no notion of equation, coefficients etc. but worked with purely geometrical quantities.

Hindu mathematicians took the Babylonian methods further so that
Brahmagupta
(598-665 AD) gives an, almost modern, method which admits negative quantities. He also used abbreviations for the unknown, usually the initial letter of a colour was used, and sometimes several different unknowns occur in a single problem.

By Midevil times Islamic mathematicians were able to discuss the importance of the unknown variable x. They were able to multiply, divide, and find the roots of polynomials and they started to put together binomial theorems. The Persian mathematician Omar Khayyam showed how to find the roots of cubic equations through line segments of intersected conic sections, but was unable to come up with an equation for cubic polynomials. In the early 13th century, however, Leonardo Fibonacci achieved a close approximation of the cubic equation: x^3+2x^2+cx=d.

In the early 16th century, Italian mathematicians Scipone del Ferro, Niccoló Tartaglia, and Gerolamo Cordano were able to solve the general cubic equation in terms of the constants in front of the variables, and Ludovico Ferrari soon found exact equations for polynomials up the fourth degree. During the 16th century it became common practice to use symbols for known and unknown algebraic powers and operations.

René Decartes, an extremely important mathematician from the 16th century, discovered analytic geometry, which reduces the solutions of geometric problems into solutions in terms of algebraic equations. He also made significant contributions to the theory of equations, including coming up with what he called "the rule of signs" for finding the positive and negative roots of equations.

For a more detailed timeline of how the history of polynomials developed, visit
http://www.vimagic.de/hope/

For more information: http://www-history.mcs.st-andrews.ac.uk/history/HistTopics/Quadratic_etc_equations.html






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Perform arithmetic operations on polynomials:

  • A-APR.1. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

Understand the relationship between zeros and factors of polynomials.

  • A-APR.2. Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x).
  • A-APR.3. Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomials.

Use polynomial identities to solve problems.

  • A-APR.4. Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x2 + y2)2 = (x2 – y2)2 + (2xy)2 can be used to generate Pythagorean triples.
  • A-APR.5. (+) Know and apply the Binomial Theorem for the expansion of (x + y)n in powers of x andy for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal’s Triangle.

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Standards from National Council of Teachers of Mathematics for grades 9 - 12:

  • All students should understand and compare the properties of classes of functions, including exponential, polynomial, rational, logarithmic, and periodic functions.



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Geogebra display of Polynomials

The quintic equation reads: a + bx + cx^2 + dx^3 + ex^4 + fx^5. Play with the coefficients to observe the effect on the graph.



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The Quintic Polynomial sketch is now hosted at GeoGebraTube. Go to the teacher's page to download, the student's page to see it in the browser.



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Wolfram|Alpha definition of a polynomial:
A polynomial is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. A polynomial in one variable (i.e., a univariate polynomial) with constant coefficients is given by:
external image NumberedEquation1.gif


This definition basically tells us that a polynomial is an expression containing a variable to a certain power multiplied by some coefficient and successive variables with their own coefficients with lower powers added to it.

Here are some examples involving polynomials of what Wolfram|Alpha can do for you.

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You can see that I typed two pretty complex polynomials into wolfram|alpha and had wolfram|alpha divide them. Wolfram|alpha then shows you a graph of the function produced, some alternate forms.



Examples of Polynomials:111parabola.png

  • x^2-x+2
  • 5k^4+3k^2+7
  • z^3+5
  • 2x-8
  • h^3-4h^2+6h-12
  • 2x+3y
  • z^2-6y

There are different degrees for these polynomials. These degrees are determined by the first exponent in the first term. For example, in the example above of x^2-x+2, the exponent on the first term is 2, therefore the degree is 2. If we know the degree we can also give it a particular name:

Degree
Name
Expression
0
Constant
7
1
Linear
x+3
2
Quadratic
x^2-x+2
3
Cubic
x^3-x^2+5
4
Quartic
6x^4-x^3+x-2

Characteristics of the degree of polynomials when graphed:

  • If the degree of a polynomial is an even number, then the limits of the graph approaching infinity and negative infinity will be going the same direction
  • If the coefficient of the highest degree term of the polynomial is positive then the limits going to the right and to the left on the graphical representation will be "up"
  • If the coefficient of the highest degree term of the polynomial is negative then the limits going to the right and to the left on the graphical representation will be "down"
  • If the degree of a polynomial is an odd number, the limits of the graph approaching infinity and negative infinity will be going in opposite directions
  • If the coefficient of the highest degree term of the polynomial is positive and the degree is an odd number, the limit of the graph going to the left will be "down" and the limit of the graph going to the right will be "up"
  • If the coefficient of the highest degree term of the polynomial is negative and the degree is an odd number, the limit of the graph going to the left will be "up" and the limit of the graph going to the right will be "down"

    For visual representations of these graphs, visit
    http://www.purplemath.com/modules/polyends.htm

Some examples of polynomials that work, and others that don't are listed below in the table:

6x^–2
This is NOT
a polynomial term...
...because the variable has a negative exponent.
1/x^2
This is NOT
a polynomial term...
...because the variable is in the denominator.
sqrt(x)
This is NOT
a polynomial term...
...because the variable is inside a radical.
4x^2
This IS a polynomial term...
...because it obeys all the rules.
Some examples and problems to work on:
  • Give the degree of the following polynomial: 2x^5 – 5x^3 – 10x + 9
This polynomial has four terms, including a fifth-degree term, a third-degree term, a first-degree term, and a constant term.
This is a fifth-degree polynomial.
  • Give the degree of the following polynomial: 7x^4 + 6x^2 + x
This polynomial has three terms, including a fourth-degree term, a second-degree term, and a first-degree term. There is no constant term.
This is a fourth-degree polynomial.


Practical Uses of Polynomials:


Higher-Level Math Classes:
Polynomials have implications to all higher-level mathematics courses. They serve as an important tool for factoring trigonometric functions, and make up the basis of the power rule in differential calculus. Mathematicians draw on different types of polynomial series to calculate slopes and mathematical approximations. Without a substantial knowledge of polynomial theory, success in any higher-level mathematics class would be very difficult.

Science:
Polynomials have relevance to nearly all the sciences. Astrophysicists use them to calculate a star's velocity and distance from another object in space. Likewise, they are important in determining pressure in applications of fluid dynamics. Chemists use polynomials to determine the composition of certain compounds and molecules, and they are central to statistics. Statistical formulas use polynomials to ascertain future values of animal birth and death rates, monetary flow and population growth.

Computers:
In the last 30 years, computer scientists have instituted important uses for polynomials. Most of their work involves locating specific targets via coordinate systems and cryptography. Polynomials are also important to travel. According to the website MathMotivation, "Without the Taylor Polynomial or other polynomial approximation, there would be no way for scientific calculators and computers to perform the calculations needed to guide our spaceships and aircraft."


Business:
Interest calculations use polynomials. If interest is compounded, then a savings account balance in the future is a polynomial of the interest rate, multiplying the balance by (1 + interest rate) for each time that the interest is compounded. A formula relating future mortgage payments to the current mortgage balance is likewise a polynomial of the interest rate, with exponents going as high as the number of future payments.

Construction:
Polynomials can be used in two-dimensional construction planning. Example (from www.experts123.com): If a certain amount of soil is available for a garden and it needs to be a certain width, the size of the garden could be calculated by using a polynomial function.

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How to Solve Higher Degree Polynomials:300px-Bring_radicals_cartoon.PNG


Solving polynomials with degrees higher than 2 can be difficult, but polynomials with some rational solutions can be solved by hand. To do this we must implement the uses of the Rational Zero Theorem and Descartes' Rule of Signs.

The Rational Zero Theorem (RZT) states: If a polynomial function, written in decreasing order, has integer coefficients, then any rational zero must be of the form ± p/q, where p is a factor of the constant term and q is a factor of the leading coefficient.

Descartes' Rule of Signs states that the Rational Zero Theorem doesn't tell us anything about what the rational zeros are, but it tells us how many zeros to expect based on the number of sign changes within the function for the positive and negative function.

Example of solving higher degree polynomials:

Consider f(x) = 6x^3+x^2-5x-2.

1) To find all the possible zeros of this function we use the RZT in the form of ± p/q, where p is -2 and q is 6.

Factors of p: ±1, ±2
Factors of q: ±1, ±2, ±3, ±6

2) We now form all the rational numbers by combining all possible values of p with all possible values of q:

When p =±1,
±p/q = ±1/1, ±1/2, ±1/3, ±1/6.

When p =±2,
±p/q = ±2/1, ±2/2, ±2/3, ±2/6.

Since ±1/1 = ±2/2, we only need to include either ±1/1 or ±2/2. The same idea applies to ±1/3 and ±2/6.

3) From our results, we can determine that all possible factors of f(x):

All possible zeros: ±1, ±2, ±1/6, ±1/3, ±1/2, ±2/3

4) Now that we have all the possible roots we must now use Descartes' Rule of Signs to determine how many positive and negative zeros there are.

For f(x) = 6x^3+x^2-5x-2 we can count how many times the function changes signs.
(Sidenote: We have to remember that when a coefficient has no negative sign, it is positive. We just don't write +1 to denote positive one.)

We can write f(x) like this: (Positive)6x^3(Positive)x^2(Negative)5x(Negative)2

The sign of f(x) changes one time so there is only one possible positive zero.

Now we can look at f(-x) to find out how many possible negative zeros there are.

We can write f(-x) like this (Negative)6x^3(Positive)x^2(Positive)5x(Negative)2

The sign of f(-x) change 2 times. Based on the rule, there is a maximum of 2 possible negative solutions. But some of the roots generated by the quadratic formula may produce complex roots. Because of this possibility, we have to count down by two's to find the complete list of possibilities. That is, based on this example, there may be two negative solutions or no negative solutions.

Since there is one positive solution, we start by exploring the values generated by the RZT. We will start with the positive 1.
To find out if 1 is a solution, we can use synthetic division. This tells us if the solution x-1 divides the function evenly.

1| 6 1 -5 -2
0 6 7 2
6 7 2 |0

As we can see, one zero of f(x) is x = 1 and we can now write the function as f(x)=(x-1)(6x^2+7x+2). Since 1 is a positive zero of f(x), and we saw earlier that there was only one possible positive zero of x, we know that x = 2, x = 1/6, x = 1/3, x = 1/2, and x = 2/3 are not zeros of f(x).
To find the last zeros, we can use the quadratic formula.

First, we can set f(x) equal to zero, since we want to find the zeros of f(x). So we have 6x^2+7x+2 = 0.
The standard form of a quadratic polynomial is ax^2+bx+c. Thus we have a = 6, b = 7, and c = 2. We can substitute these values into the quadratic formula:

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x = -7±sqrt[7^2-4(6)(2)]
2(6)

= -7±sqrt(49-48)
12

= -7±1
12

Thus, we can see that x = -8/12 = -2/3 or x = -6/12 = -1/2.

Therefore we know the zeros of f(x) are x = 1, x = -2/3 and x = -1/2.



Solving Polynomials Through Long Division and Synthetic Division


When setting up polynomial division, first, set up the division like this:
Picture_5.png
Place the polynomial that you are dividing on the inside and what you are dividing that polynomial by on the outside.

Look at the leading x of the divisor and the x^2 of the dividend. If you divide the leading x^2 on the inside by the leading x of the divisor, what would you get? You'd get an x. Another way to think of this: what can you multiply the leading x by to get the leading x^2 in the dividend? So, we put and x on top like this:
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Next, take that x and multiply it through the divisor, placing the result underneath like this:
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We want to subtract this polynomial from the dividend, so we draw a bar underneath and subtract the elements of the second polynomial:
Picture_9.png
We also have to remember to bring down that -10 from the dividend, (like we are subtracting nothing from -10):
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Next, we want to compare the leading x of the divisor to the remaining -10x. If we divide -10x by x, what do we get? SInce the answer is -10, we put a -10 on top:
Picture_11.png
Just like before, multiply the divisor by the -10 and put result underneath like so:
Picture_12.png
Then draw a bar and subtract:
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This means that (x^2-9x-10)/(x+1)=x-10.
Because we were left with 0 at the bottom, our equation has no remainder, but this is not always the case. If there were a remainder that did not divide evenly into x, we would take that remainder, divide it by the divisor, and place it on top with the rest of the solution.

Here is another example of long division of polynomials:
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Long division of polynomials is important because it provides us with a way to factor higher degree polynomials into smaller polynomials that we can work with more easily.

How to solve 3^2+8^2+(-9)x+2/(x-1):


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Factoring Polynomials Flowchart

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Youtube Video on Polynomials:




Binomial Expansion Tool


The formal expression on the Binomial Theorem is as follows:

Binomial.gif

However, this formula means very little to higher level mathematicians, let alone high school students. Let's look at how to apply this theorem to binomial expansion.
Recall that

Combination.gif

where "n!" means "the product of all the whole numbers between 1 and n."

Here is an example of how combinations works:

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Combinations are a key part of binomial expansion. Here are some examples of binomial expansion at work:

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Expansion.png

This is a Geogebra tool that looks at binomial expansions, to help illustrate the relationship between that process and Pascal's Triangle.

BinomialExpansion.png


The Binomial Expansion sketch is now hosted at GeoGebraTube. Go to the teacher's page to download, the student's page to see in the browser.

History of Polynomial Resources
- http://www.algebra.com/algebra/about/history/
- http://www.vimagic.de/hope/

Wolfram|alpha Resources
http://www.wolframalpha.com/input/?i=polynomial

Examples of Polynomials Resources
- http://www.mathsisfun.com/algebra/degree-expression.html
- http://www.purplemath.com/modules/polydefs
- http://www.purplemath.com/modules/polyends.htm

- <a title='By Jamme123 (Own work) [CC-BY-3.0 (www.creativecommons.org/licenses/by/3.0)], via Wikimedia Commons' href='http://commons.wikimedia.org/wiki/File:Sextic_Graph.png'><img width='512' alt='Sextic Graph' src='http://upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Sextic_Graph.png/512px-Sextic_Graph.png'/></a>

Practical Applications of Polynomials Resources
- http://www.ehow.com/about_5438361_uses-polynomials.html#ixzz1FHZ9jU2Q
- http://www.instructables.com/id/How-to-factor/step8/Factoring-polynomials-by-synthetic-division/
- http://www.freeclipartnow.com/construction/workers/construction-kid.jpg.html

Polynomial Division
- http://www.purplemath.com/modules/polydiv2.htm


Factoring Polynomials
- http://www.schools.manatee.k12.fl.us/072JWILLIAMS/072jwilliams1/factoring_polynomials_flowchart.html

Binomial Expansion
- http://www.purplemath.com/modules/binomial.htm
http://www.purplemath.com/modules/binomial.htm