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A detailed tutorial on how to use algebra tiles. Step by step tutorial including several examples of how to use algebra tiles for reference.

Algebra tiles are a visual expression of polynomials and polynomial equations. Each tile is meant to represent a different polynomial. A large square tile represents the squared variable, a smaller square tile represents a single number, with no variable, and a rectangle represents the single variable. The tiles are red and green. Green represents positive monomials, and red represents negative monomials. Tiles can be combined to create equations, or the same tiles can be combined to express the coefficient. Addition and subtraction can be performed by adding and removing tiles.

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A detailed tutorial on degrees of polynomials. Step by step tutorial including several examples of degrees of polynomials for reference.

The degree of a polynomial is the highest power found in it. For example, in your normal quadratic equation, the degree is two, because the highest power – the highest number found in an exponent – is a two. In other polynomials, the degree may be something different. No matter what order the variables and their powers are placed in, the degree is always the highest one. For example. the degree of x^2 + x + 7 is exactly the same as x + 7 + x^2.

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A detailed tutorial on identifying zero polynomials. Step by step tutorial including several examples of identifying zero polynomials for reference.

A zero polynomial is the additive identity of an additive group of polynomials. So this means it is not a unique polynomial, even though it may seem like it. In order to identify a zero polynomial, you need to be aware of the two properties that zero polynomials possess. The first one is that all coefficients of a zero polynomial are zero, and add up to zero. The second is that a zero polynomial doesn’t have a degree – it is an undefined degree. Typically people will write this as a degree of -1, or more common, of negative infinity.

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A detailed tutorial on invariants and the property of invariance. Step by step tutorial including several examples of invariants for reference.

Invariants are any function or number that displays the property of invariance. Invariance is when a function or number can go through several transformations without changing, or without going outside of its set parameters. The set parameters differ depending on the function or number. Some examples of invariant functions and numbers are the absolute value of a complex number, the degree of a polynomial, and certain parts of a square matrix

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A detailed tutorial on how to identify prime polynomials. Step by step tutorial including several examples of identifying prime polynomials for reference.

Prime polynomials are any polynomial that cannot be factored. Just like a number is prime if you can not break it down into two seperate whole numbers to multiply, a polynomial is prime if you cannot break it down into two separate binomials with whole numbers to multiply. When you run into a prime polynomial when trying to solve a quadratic equation, you cannot use the factoring method. what the factoring method does is split the polynomials into a binomial, which cannot be done to a prime polynomial. If you have a prime polynomial, you have to use the quadratic formula to solve it. At first, you can spot prime polynomials by attempting to factor it, but eventually you will be able to do it just by looking at it.

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A detailed tutorial on Mandelbrot sets and identifying Mandelbrot sets. Step by step tutorial including a several visual examples of a Mandelbrot set for reference.

A Mandelbrot set is defined as a set of points in the complex frame, the boundary of which forms a fractal. This can be mathematically defined as the set of complex values c for which the orbit of zero under iteration of a complex quadratic polynomial remains bounded.

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A detailed tutorial on how to find oblique asymptotes. Step by step tutorial including several examples of how to find oblique asymptotes for reference.

There are several different types of asymptotes. In this tutorial, we will be discussing oblique asymptotes. In order to find the oblique asymptotes of a function, you must first determine if the asymptote slants. If the numerator of a rational function has exactly one degree greater than the denominator, then the function slants and therefore has an oblique asymptote. When you divide the numerator and the denominator, the term or polynomial you get is the oblique asymptote.

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A detailed tutorial on polynomial long division. Step by step tutorial including several examples of polynomial long division for reference.

Polynomial long division is a mix of regular long division and rules of polynomials – it looks confusing at first, but isn’t too difficult to follow. Polynomial long division is actually a type of algorithm. It is only used when dividing a polynomial by another polynomial of either the same or a lower degree. The “degree” of a polynomial is the highest power in the polynomial, and the terms in the polynomial should be ordered from highest degree to lowest degree. When using polynomial long division, you must write out all coefficients and terms, even “invisible” ones – ones that have a coefficient of zero and so are typically not written in the polynomial. Polynomial long division is solved the same way as regular long division

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A detailed tutorial on determining roots of polynomials using Descartes’ rule of signs. Step by step tutorial including several examples of how to determine roots of polynomials using Descartes’ rule of signs for reference.

Descartes’ rule of signs is used to determine the number of positive or negative roots in a polynomial. It does not give the exact number of roots, but it does give a close estimate. There are two versions of this rule – one for positive roots, and one for negative roots. The rule for positive roots states that the number of positive roots of the polynomial is either equal to the number of sign differences between consecutive nonzero coefficients, or less than it by a multiple of two. The rule for negative roots states that the number of negative roots is the number of sign changes after negating the coefficients of odd power terms, or less than it by a multiple of two.