This document provides solved examples on polynomials, including determining if expressions are polynomials, writing polynomials in standard form, finding the degree of polynomials, and classifying polynomials as monomials, binomials, trinomials, or by degree. The examples cover concepts such as the definition of a polynomial, degrees of polynomials, and identifying coefficients and terms within polynomials. Step-by-step solutions demonstrate how to analyze and classify different types of polynomial expressions.

1. MathBuster Solved Examples
Lesson 02 Title: Polynomials
Assignment Code: MB0902.4.1
The solved examples in this assignment are based on the following concepts:
1. Definition of a polynomial in one variable, its coefficients, with examples and counter
examples, its terms, zero polynomial.
2. Degree of a polynomial. Constant, linear, quadratic, cubic polynomials;
3. Monomials, binomials, trinomials.
Example MB0902.4.1.01
Which of the following algebraic expressions are polynomials in one variable? Give reasons
for your answers.
a. –
b.
c.
d.
MathBuster Solution Strategy
For each of the given expression we need to check two properties:
1. Is it a polynomial?
2. Is it a polynomial in one variable?
Polynomials are special types of algebraic expressions which satisfy 2 conditions:
1. The denominator of a polynomial is always 1.
2. The power of each variable must be a non-negative integer.
Thus, for each of the given expressions, we need to check the above two conditions. In
addition, we also will need to check if the expressions contain only one variable.

2. Answer MB0902.4.1.01a:
The given expression is – does not have a variable quantity in the denominator.
We can write,
–
– .
Thus, the first condition of it being a polynomial is satisfied.
Now, there are three terms in this expression, . Thus, the power of the
variable in the three terms is 4, 1 and 0. Since none of these is negative, the expression
satisfies the second condition of being a polynomial.
Therefore, the expression – is a polynomial.
Now, we observe how many variables are there in this expression. Since there is only ONE
variable, x, we conclude that – is a polynomial is one variable.
Answer MB0902.4.1.01b:
The given expression, has two terms, .
In the second term, the variable x is in the denominator. Using the rules of exponents, we can
write,
.
We can see that this term contains a negative power – of the variable. Thus, the second
condition of being a polynomial is not met by this expression.
Therefore, is not a polynomial.
Answer MB0902.4.1.01c:
The given expression, has no variable in the denominator of this expression
and we can write this as

3. Thus, the first condition of it being a polynomial is satisfied. Now, let’s look at each term of
the expression. The given expression, has three terms, .
Now, we can write
.
We can see that this term contains a fractional power of the variable. Thus, the second
condition of being a polynomial is not met by this expression. Therefore, is
not a polynomial.
Answer MB0902.4.1.01d:
The given expression is does not have a variable quantity in the denominator. We can
write,
.
Thus, the first condition of it being a polynomial is satisfied.
Now, there are three terms in this expression, . Thus, the power of the variable
in the each of the two terms is 1. Therefore, the expression satisfies the second condition of
being a polynomial. Therefore, the expression is a polynomial.
Now, we observe how many variables are there in this expression. Since there are TWO
variables (y and z), we conclude that is NOT a polynomial in one variable.
Example MB0902.4.1.02
Write the coefficients of x3 in the following polynomials:
a. –
b. –
c.
d. –
e. –

4. MathBuster Solution Strategy
The coefficient of a term in a polynomial is the constant factor of the term. The for the term
axn, the coefficient is a.
Thus, for each polynomial given above, we must:
(a) Identify the term that contains x3.
(b) Identify the constant factor of this term.
Answer MB0902.4.1.02:
The following table provides us with the coefficients for each polynomial.
Polynomial Term containing x3 Coefficient of x3
1
There is NO term containing .
There is NO term containing .
Example MB0902.4.1.03
Write each polynomial in standard form and the determine the degree of the polynomial:
a. –
b. –
c.
d. –
e. –

5. MathBuster Solution Strategy
To write a polynomial is standard form, we must combine all like terms and then arrange the
terms in decreasing powers of variables. The degree of a polynomial on one variable is the
highest power of the variable in each of its terms, after the polynomial has been written in
standard form.
Thus, to determine the degree, we must:
a. Combine all like terms of the polynomial.
b. Arrange them in decreasing order of powers.
c. Observe the highest power of the variable.
Answer MB0902.4.1.03a:
The given polynomial is – . Observe that all like terms are already combined.
Rearranging the terms so that they are in decreasing power of the variable, we get
The powers of the variables in different terms are 3, 2 and 0. The highest power of the
variable is 3. Therefore, the degree of the polynomial is 3.
Answer MB0902.4.1.03b:
The given polynomial is – .
Rearranging the terms to bring all the like terms together, we get
– .
Combining all the like terms, we get
– .
The terms are already arranged in decreasing powers of the variable. The powers of the
variables in different terms are 4, 3, 2 and 0. The highest power of the variable is 4.
Therefore, the degree of the polynomial is 4.
Answer MB0902.4.1.03c:
The given polynomial is
Rearranging the terms to bring all the like terms together, we get
Combining all the like terms, we get

6. The terms are already arranged in decreasing powers of the variable. The powers of the
variables in different terms are 3 and 0. The highest power of the variable is 3. Therefore, the
degree of the polynomial is 3.
Answer MB0902.4.1.03d:
The given polynomial is – . In this case, all the like terms are already combined and they
are arranged in decreasing order of the powers of the variable. The terms are already arranged
in decreasing powers of the variable. The powers of the variables in different terms are 2, 1
and 0. The highest power of the variable is 2. Therefore, the degree of the polynomial is 2.
Answer MB0902.4.1.03e:
The given polynomial is – . Rearranging the terms to bring all the
like terms together, we get
Combining all the like terms, we get
The terms are already arranged in decreasing powers of the variable. The powers of the
variables in different terms are 2 and 1. The highest power of the variable is 2. Therefore, the
degree of the polynomial is 2.
Example MB0902.4.1.04
Classify the following polynomials on the basis of the number of terms they have:
a. –
b. –
c.
d. –
e. –
MathBuster Solution Strategy
Polynomials have special names based on the number of unlike terms they have.
Monomials are polynomials with ONE term.
Binomials are polynomials with TWO terms.
Trinomials are polynomials with THREE terms.

7. Answer MB0902.4.1.04:
The following table provides classification of the polynomials given.
Polynomial Like Terms Combined Number of Terms Name
3 Trinomial
3 Trinomial
2 Binomial
1 Monomial
2 Binomial
Example MB0902.4.1.05
Classify the following polynomials on the basis of their degree:
a. –
b. –
c.
d. –
e. –
MathBuster Solution Strategy
Polynomials have special names based on their degree.
 Polynomials of degree ZERO are called Constant Polynomials.
 Polynomials of degree ONE are called Linear Polynomials.
 Polynomials of degree TWO are called Quadratic Polynomials.
 Polynomials of degree THREE are called Cubic Polynomials.
Answer MB0902.4.1.05:
The following table provides classification of the polynomials given.
Polynomial Like Terms Combined Degree Name
3 Cubic
2 Quadratic

8. 3 Cubic
2 Quadratic
1 Linear
3 0 Constant
Next Step Recommended:
To see how well you have understood these problems, take the quiz MB0902.TQ.1
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