Unit Content
Polynomial: Is an algebraic expression consisting of a sum of two or more monomials. A monomial is a product of a “coefficient” or “constant”, and a power of a variable. The coefficient may be a specific number, or be denoted by a letter from the beginning of the alphabet: a, b, c, etc. Even when the coefficient is designated by a letter, it is considered a “constant”, which means it is treated as being known for purposes of the calculation to be done. A variable is a number that is not known, at least at the start of the problem, but which can be taken from some collection of numbers (whole numbers, integers, rational numbers, real numbers, etc., etc.). A variable is usually denoted by a letter from the end of the alphabet: x, y, z, etc. A power of a variable is the product of a certain number of copies of the variable, with the number of copies indicated by an exponent. Thus x2 = x ´ x, and x 3 = x ´ x ´ x, and so forth. The fact that these products do not depend on how the copies of the variable are multiplied together, only on the total number of copies, is fundamental, and is a consequence of the Associative Rule for Multiplication. This will be discussed in further detail below. Polynomials can be combined using the basic mathematical operations, addition, subtraction, and multiplication. Dividing one polynomial by another usually results in a more general type of expression, often called a rational function. For this reason, we will avoid division of polynomials in this unit.
A few of the commonly used polynomials are:
Monomial: By monomial, we mean a product of axn, where a indicates a constant (any known number), x is a variable, and xn is a power of x, and where n = 0 is a non-negative whole number.
For example, the monomial 2x3, is interpreted as 2 copies of x to the 3rd power.
Binomial: polynomial consisting of two monomial terms
Trinomial: polynomial consisting of 3 monomial terms
The exponent of x in a monomial axn is called a degree. The largest degree of the monomials in a polynomial is called the order of the polynomial. Order allows you to identify the type of parent function or polynomial you are working with. In the 8th grade we are only required to work with polynomials of the third order or less in a variety of ways. While there are higher order polynomials, they are often very difficult to work with and they are often unable to be factored effectively once they reach the fifth order (x 5).
3x 2 + 4 x – 2
The above polynomial is an example of a trinomial; containing 3 terms of degrees 0, 1, and 2. The highest degree is 2, so this is an order 2 polynomial, often also called a quadratic polynomial. There are a few other specific conditions for defining a polynomial that are not important in this unit but can be mentioned, for example: all exponents must be non-negative whole numbers (including zero).
Area Model: An area model represents a multiplication by the area of a rectangle, subdivided into sub-rectangles whose areas are the products of the place value pieces of the factors. The area model provides an effective method for helping students understand multiplication of two-digit numbers, but beyond that, it becomes rather unwieldy and not really practical. However, a symbolic version of the area model is very useful for understanding the extended distributive rule, and I will discuss this symbolic version with my students.
The Box Method: A modification of the area model often known as the box method, is a rectangular model used in math for multiplication. In the box method, one makes an array, with one column for each place value part of the first factor, and one row for each place value part of the second factor. Then in the box corresponding to a given pair of place value parts, one from each factor, one puts the product of the two parts. The whole product is found by summing all these products. The lattice method can be seen as a slightly refined and more automated, but less transparent, version of the box method.
Figure 3: Box Method Model (2 × 2)
We will give examples of the Box Method below.
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