Section P.3: Introduction to Polynomials
Chapter P – Polynomials
Definition of a monomial:
A monomial is a variable, a real number, or a multiplication of one or more variables and a real number
with whole-number exponents. A monomial may contain a numerical fraction, but it may not have a
variable in the denominator of a fraction.
#1-16: Classify the following terms as monomials or not monomials
1) 5x3
Solution: this is a monomial
3) 3x-4
Solution: this is not a monomial (because of the negative exponent)
5) -y6
Solution: this is a monomial
7) -11xy3
Solution: this is a monomial (it is okay to have more than one letter)
9)
2
𝑎𝑏 2 𝑐
3
Solution: this is a monomial (it is okay to have a number as a fraction, you just can’t have a letter in
the denominator of a fraction)
11)
1
𝑦𝑥 −4
2
Solution: this is not a monomial because of the exponent of (-4)
13)
2
5𝑥
Solution: this is not a monomial (because of the x in the denominator of a fraction)
15)
2
𝑥+5
Solution: this is not a monomial (because of the x in the denominator of a fraction)
#17 – 28: Determine the degree and coefficient of each of the following monomials
17) 8x2
Solution: this is a 2nd degree polynomial. The coefficient is the 8.
19) z3
I think of this as 1z3
Solution: This is a 3rd degree polynomial with coefficient of 1.
21)
2
𝑥
3
𝟐
Solution: This is a 1st degree polynomial with coefficient of 𝟑.
23) 3x2y6
I need to add the exponents of the x and the y to get the degree of the term.
Solution: The degree is 8. The coefficient is 3.
25) -x4y5
Think of this as -1x4y5
I need to add the exponents of the x and y to get the degree.
Solution: degree 9, coefficient -1
27) -xyz
Think of this as -1x1y1z1
Add the exponents of the x, y, z to get the degree
Solution: degree 3, coefficient -1
Polynomial definition:
A polynomial is a monomial or the sum or difference of monomials. Each monomial is called a term of
the polynomial.
Important!:Terms are separated by addition signs and subtraction signs, but never by multiplication
signs
A polynomial with one term is called a monomial
A polynomial with two terms is called a binomial
A polynomial with three terms is called a trinomial
#29-44: Classify the following terms as polynomials or not polynomials. If the expression is a
polynomial classify it as a monomial, binomial, trinomial or other.
29) 4x + 5
Solution: this is a polynomial it is also a binomial
31) x2 + 5x +6
Solution: this is a polynomial it is also a trinomial
33) 3x-4
Solution: this is not a polynomial because of the negative exponent
5
35) x3 - 𝑥
Solution: this is not a polynomial because of the x in the denominator of a fraction
37)
𝑥+3
𝑥−4
Solution: this is not a polynomial because of the x in the denominator of a fraction
39) xy3 + 4x + 5y –7
Solution: this is a polynomial (but it is not a monomial / binomial nor trinomial) thus it is an other
polynomial
41) 5x4 + 3xy
Solution: this is a polynomial it is also a binomial
43) 5 +
3
𝑥+𝑦
Solution: this is not a polynomial because of the x and y in the denominator of a fraction
The degree of a polynomial is the highest of the degrees of all its terms.
The leading term of a polynomial is the term with the highest degree
The leading coefficient of a polynomial is the coefficient of the term with the highest degree.
#45-56: Find the leading term of each polynomial, then state the degree of the polynomial and the
leading coefficient.
45) 5x3 + 3y
Solution: the 5x3 is the leading term, the polynomial is degree 3, the leading coefficient is 5.
47) 3x3 + 5xyz3
The 3x3 term has degree 3, while the 5xyz3 term has degree 5 (add the exponents on each of the letters)
Solution: the 5xyz3 is the leading term (because it has the highest degree), the polynomial is degree 5,
the leading coefficient is 5
49) 8x + 5x2 – x4
The –x4 (think of this as -1x4) is the leading term as it has the degree 4 which is higher than the rest of
the terms.
Solution: -x4 is the leading term, the polynomial has degree 4, the leading coefficient is -1
51) -xy5 + 2yz – 3x2y
If you add the exponents of the letters, the –xy5 (think of this as -1xy5) term has degree 6, which is the
highest of all the terms.
Solution: -xy5 is the leading term, the polynomial has degree 6, the leading coefficient is -1
53) 7
If I need to, I can think of this as 7x0 as 7x0 = 7*1 = 7
Solution: the 7 is the only term, thus it is the leading term. The polynomial is degree 0, the leading
coefficient is 7.
55)
−1
2
Solution: the
coefficient is
−1
is
2
−1
.
2
the only term, thus it is the leading term. The polynomial is degree 0, the leading
#57 – 68: Evaluate each polynomial using x = 2, y = -3 and z = 4
57) 5x3 + 3y
= 5(2)3 + 3(-3)
= 5(8) + 3(-3)
= 40 – 9
Solution 31 (it would be wrong to write x = 31, as x equals 2 in this problem)
59) 3x3 + 5xyz3
=3(2)3 + 5(2)(-3)(4)3
= 3(8) + 5(2)(-3)(64)
= 24 – 1920
Solution: -1896
61) 8x + 5x2 – x4
8(2) + 5(2)2 – (2)4
8(2) + 5(4) – 16
16 + 20 – 16
Solution 20
63) -xy5 + 2yz – 3x2y
I might think of this as -1xy5 + 2yz – 3x2y
= -1(2)(-3)5 + 2(-3)(4) – 3(2)2(-3)
= -1(2)(-243) + 2(-3)(4) – 3(4)(-3)
= 486 – 24 + 36
Solution: 498
65) 7
Solution 7 (this is a constant function, since it doesn’t have any letters and it will always equal 7
regardless of the values of the letters)
67)
−1
2
Solution
−𝟏
𝟐
(this is a constant function, since it doesn’t have any letters and it will always equal 7
regardless of the values of the letters)