In previous chapters, when you have solved equations, you have put to work a variety of tools that have allowed you to manipulate the expressions and terms equations contain. These tools can be brought forward and extended as you solve polynomial equations. Table 8.1 lists a few of the fundamental concepts you work with as you solve polynomial equations. Subsequent sections of this chapter elaborate on these concepts.
Table 8.1. Concepts Related to Solving Polynomial Equations
Practice
Discussion
Addition
The addition property allows that if you begin with an equation such as a = b, then you can add equal values to both sides of the equation: a + c = b + c.
Multiplication
The multiplication property allows that if you begin with an equation such as a = b, then you can multiply both sides of the equation by equal values to obtain an equivalent equation: a × c = b × c. When you multiply the sides of an equation by an expression that contains a variable, you must take a few precautions. One involves checking whether you are multiplying by a value that is equal to zero.
Coefficients
With a term such as 3a2, the constant (3) preceding the variable a is the coefficient of the term. In a polynomial, you often find several coefficients for any given variable: 4b2 + 3b3 + 4b4. In this polynomial, you find three coefficients for three separate instances of the variable b.
Like Terms
If you find terms that are raised to the same power, such terms are like terms. You can also refer to them as similar terms.
Degrees
The degree of a term is the value of its exponent. For the term 3x5, the variable x possesses an exponent of 5, and this is the degree of the term. For the term 3x, the degree is 1. Each term of a polynomial that contains a variable also possesses a degree. When a polynomial contains several terms, each with a different degree, then the degree of the polynomial is the largest degree of the constituent terms. For example, with 3x2 + 4x3 + 3x4, the degree of the polynomial is 4, because the highest degree of the terms is 4.
Order
You can organize the terms in a polynomial according to ascending or descending order. To organize the terms of a polynomial in ascending order, begin on the left with the term that contains the smallest exponent (3x2 + 4x3 + 3x4). To organize the terms of a polynomial in descending order, begin on the left with the term that contains the largest exponent (3x4 + 4x3 + 3x2).
Zero Products
If you begin with two numbers, and one of the numbers is equal to 0, then the product is zero. Along the same lines, if you start with either a = 0 or b = 0, then you can create an equivalent equation that reads ab = 0.
Missing Terms
If you want to write expressions in standard ways, you can use the coefficient 0 to identify terms that are absent or missing from a polynomial. For example, you can rewrite a2 + a4 + a5 as a2 + 0a3 + a4 + a5.
Classification
A polynomial consisting of one term is a monomial. A polynomial consisting of two terms is a binomial. A polynomial consisting of three terms is a trinomial.
