Interactive Algebra Review

Based on the algebra reviews in Applied Calculus and Finite Mathematics and Applied Calculus

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5. Solving Polynomial Equations
Part A: Equations

Note To review factoring, go back to 3B: Multiplying Algebraic Expressions. If you are comfortable with the notion of what an equation is, go on to Part B: Solving Polynomial Equations.

Q What is an equation?


An equation is the statement that two mathematical expressions are equal. In other words, it consists of two mathematical expressions separated by an equal sign.

The letters that occur in an equation signify numbers. Some stand for well-known numbers, such as , c (the speed of light: 3108m/sec) or e (the base of natural logarithms: 2.71828 . . .). Some stand for variables or unknowns. Variables are quantities (such as length, height, or number of items) that can have many possible values, while unknowns are quantities whose values you may be asked to determine. The distinction between variables and unknowns is fuzzy, and mathematicians often use these terms interchangeably.

A solution to an equation in one or more unknowns is an assignment of numerical values to each of the unknowns, so that when these values are substituted for the unknowns, the equation becomes a true statement about numbers.

Example of an Equation

    x + y = 7

is a linear equation in two unknowns, x and y. A solution to this equation is x = 2, y = 5, or (2, 5), since substituting 2 for x and 5 for y yields the true statement

    2 + 5 = 7.

Other solutions are (0, 7), (0.5, 6.5), and (-2, 9).

We could also think of x + y = 7 as an equation in two variables, as the numbers x and y could stand for quantities that can vary. For example, x could stand for the number of days per week you attend math class and y for the number of days per week you don't attend math class. The equation x + y = 7 then amounts to the statement that there are a total of seven days in the week. If you knew the number x, you could find the remaining unknown, y.

An equation in one unknown has exactly one variable, and the Courier x is traditionally reserved for that purpose (like most traditions, it is not strictly followed)

Examples Equations in One Unknown

Here are a few equations in one unknown:

    3x + 4 = 0
    x2 - 3x + 2 = 0
    4x4 + 11x2 + 9 = 0
    x5 - 10x + 5 = 0 x0.5 -2x2 = 4x

Here are some for you.

There are two methods of solving an equation: analytical and numerical. To solve an equation analytically means to obtain exact solutions using algebraic rules. To solve it numerically means to use a computer or a graphing calculator to obtain solutions. In this appendix, we shall concentrate on the analytical approach.

We should point out that almost anything can happen when you try to solve an equation. Here are the possibilities, illustrated by examples.

Solutions of Equations

1. Unique Solution
This means that the equation has one, and only one, solution.

The equation 3x + 12 = 0 has the unique solution x =      

Sometimes the solution is not so easy to find. Often, it cannot be found at all analytically. An example is x5-10x+10 = 0, whose unique real solution can only be found numerically.

2. Two or More Solutions
An equation can often have more than one solution.

The equation x2 - 3x + 2 = 0 has the following two solutions

x =   and x =      

Just as in the case of a unique solution, multiple solutions may not be easy to find. An example is x5-10x+5 = 0, whose three real solutions can only be found numerically.

3. No Solutions
The equation 4x4 + 11x2 + 9 = 0 has no real solutions whatever. Think for a while about why this should be the case before clicking here.


Mentally solve the following equations for x. (That is, try to solve them by writing down as little as possible.)

x - 3 = 1has the unique solution x =  
3x + 1 = xhas the unique solution x =  
x + 1 = 3x + 1has the unique solution x =  

+ 1 = 3
has the unique solution x =  

x - 1
= 5
has the unique solution x =  
ax = 1 (a 0)has the unique solution x =  
ax + b = c (a 0)has the unique solution x =  

By factoring the left-hand side (or any other method) find all possible solutions for x in the equations in the next group of exercises. If there are two solutions, enter each solution in a different box. If there is only one solution, enter it in the first box, and leave the other box blank. Try to work mentally as far as possible. When you do decide to resort to pen and paper, be as meticulous as possible about the way you write mathematics, using as many steps as you like. This is the key to communicating your train of thought to the people assessing your work (they like that). Equally important, it helps you organize your thoughts more clearly.

x2 - x = 0 has solution(s)  
2x2 = 2 has solution(s)  
x2 - 5x + 6 = 0 has solution(s)  
x2 + 2x + 1 = 0 has solution(s)  

The above tutorial covers only a portion Section A.5 of the Algebra Review in Applied Calculus and Finite Mathematics and Applied Calculus. Before attempting all the exercises there, we suggest you first go on to Part B of this tutorial: solving polynomial equations.

Last Updated: March, 2006
Copyright © 2001 Stefan Waner and Steven R. Costenoble

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