Chapter 2 Theory of a polynomial Equations
Chapter 2
Theory of a Polynomial Equation
2.1 A polynomial Equation
One main goal in algebra is to keep expanding our knowledge of solving equations. In this
chapter we will learn several facts that are useful in solving a polynomial equation. As we
stated before the polynomial equation is a polynomial function set equal to zero,
.
Definition (A polynomial equation )
The general form of a polynomial equation of degree in the unknown variable is
Where are known coefficients (numbers), , and must be a
positive integer.
As you may have discovered, algebra class is not the only place to use polynomial equations.
In fact, polynomial equations are used in many fields, including problems in engineering,
biology, economics, physics, and many more. You have already done a lot of work with
polynomial equations of degree 1 and of degree 2. However, if you were doing research in
one of the fields mentioned above, the equations you would work with might be of higher
degree. In this chapter, you will learn some theorems and techniques that you can use to
solve higher degree polynomial equations. This study is called the “theory of equations.”
2.2 Evaluating a Polynomial
Recall that a polynomial function can be evaluated at a number in its domain by substituting
that number for the variable and then performing the indicated operations. For example, to
evaluate the polynomial – at , substitute 2 into the polynomial
function to obtain – There are other ways to evaluate
polynomials. In this section you will learn how to use synthetic division and the Remainder
Theorem to evaluate a polynomial.
2.3 Polynomial Long Division
Before learning about synthetic division, here is a review of polynomial long division. To
divide one polynomial by another using polynomial long division, write the problem in long
division form then perform division. See the following example.
10
,Chapter 2 Theory of a polynomial Equations
Example 1 (Long division)
Use long division to perform the division
Solution
That is,
You can rewrite the result of this division in another form by multiplying both sides of the
equation by the divisor, :
– –
In arithmetic, we write
Equivalently,
and is called the divisor, is the dividend, is the quotient, and is the remainder.
That is
11
, Chapter 2 Theory of a polynomial Equations
Dividend Remainder
= Quotient +
Divisor Divisor
or
Dividend = Divisor · Quotient + Remainder
In algebra, if we divide a polynomial by a polynomial (where the degree
of is less than the degree of , we would find
The polynomial (numerator polynomial) is called the dividend, is the divisor,
as the quotient, and is the remainder.
If you divide the polynomial by the binomial – then
– .
Here is a fixed number, is a polynomial, and is a number. For example, if we
divide by , using division , we find (try to prove)
or
That is,
dividend
divisor
Quotient
Remainder
2.4 Synthetic Division
Synthetic division is a shorthand method for dividing a polynomial by a binomial of
the form – .
To divide a polynomial by a binomial of the form – using synthetic division:
1. Set up the synthetic division as follows: draw a half box and write the number in it.
To the right, write the coefficients of the terms of the dividend in descending order,
using 0 whenever a power is missing. Draw a line below these coefficients.
2. Starting on the left, bring the first coefficient down below the line.
12
Chapter 2
Theory of a Polynomial Equation
2.1 A polynomial Equation
One main goal in algebra is to keep expanding our knowledge of solving equations. In this
chapter we will learn several facts that are useful in solving a polynomial equation. As we
stated before the polynomial equation is a polynomial function set equal to zero,
.
Definition (A polynomial equation )
The general form of a polynomial equation of degree in the unknown variable is
Where are known coefficients (numbers), , and must be a
positive integer.
As you may have discovered, algebra class is not the only place to use polynomial equations.
In fact, polynomial equations are used in many fields, including problems in engineering,
biology, economics, physics, and many more. You have already done a lot of work with
polynomial equations of degree 1 and of degree 2. However, if you were doing research in
one of the fields mentioned above, the equations you would work with might be of higher
degree. In this chapter, you will learn some theorems and techniques that you can use to
solve higher degree polynomial equations. This study is called the “theory of equations.”
2.2 Evaluating a Polynomial
Recall that a polynomial function can be evaluated at a number in its domain by substituting
that number for the variable and then performing the indicated operations. For example, to
evaluate the polynomial – at , substitute 2 into the polynomial
function to obtain – There are other ways to evaluate
polynomials. In this section you will learn how to use synthetic division and the Remainder
Theorem to evaluate a polynomial.
2.3 Polynomial Long Division
Before learning about synthetic division, here is a review of polynomial long division. To
divide one polynomial by another using polynomial long division, write the problem in long
division form then perform division. See the following example.
10
,Chapter 2 Theory of a polynomial Equations
Example 1 (Long division)
Use long division to perform the division
Solution
That is,
You can rewrite the result of this division in another form by multiplying both sides of the
equation by the divisor, :
– –
In arithmetic, we write
Equivalently,
and is called the divisor, is the dividend, is the quotient, and is the remainder.
That is
11
, Chapter 2 Theory of a polynomial Equations
Dividend Remainder
= Quotient +
Divisor Divisor
or
Dividend = Divisor · Quotient + Remainder
In algebra, if we divide a polynomial by a polynomial (where the degree
of is less than the degree of , we would find
The polynomial (numerator polynomial) is called the dividend, is the divisor,
as the quotient, and is the remainder.
If you divide the polynomial by the binomial – then
– .
Here is a fixed number, is a polynomial, and is a number. For example, if we
divide by , using division , we find (try to prove)
or
That is,
dividend
divisor
Quotient
Remainder
2.4 Synthetic Division
Synthetic division is a shorthand method for dividing a polynomial by a binomial of
the form – .
To divide a polynomial by a binomial of the form – using synthetic division:
1. Set up the synthetic division as follows: draw a half box and write the number in it.
To the right, write the coefficients of the terms of the dividend in descending order,
using 0 whenever a power is missing. Draw a line below these coefficients.
2. Starting on the left, bring the first coefficient down below the line.
12
