4.1 Polynomial Terminology and Adding and Subtracting Polynomials

Leanne Thompson

4.1 Polynomial Terminology and Adding and Subtracting Polynomials

Polynomials are a type of algebraic expression that consist of numbers, variables, and/or exponents. We will begin studying polynomials by covering some important terminology.

Terminology

The table below reviews some important terminology related to polynomials.

Table 4.1.1
Term and Definition	Examples
A variable is a letter that represents an unknown number or numbers.	In the expression y²+ y + 6, the variable is y.
A polynomial is an algebraic expression that consists of numbers, variables, exponents, and arithmetic operations such as +, – , $\times$ , and $\div$ . In a polynomial, the exponents of the variables must be non-negative integers.	5x + 2 3a – (4b + 6) x²y²+ xy – 7
A term is a number or the product of a number and one or more variables. Terms in a polynomial are separated by addition or subtraction signs.	In the expression 3x – 13y²+ 73xy, there are three terms: 3x, –13y², and 73xy.
A monomial is a polynomial with exactly one term.	$8$ $-2{x}^{2}$
A binomial is a polynomial with exactly two terms.	$27{z}^{3}-8$ $13-5{m}^{3}$
A trinomial is a polynomial with exactly three terms.	$4{y}^{2}-8y-6$ $-12{m}^{3}-5{m}^{2}-2m$
A constant term is a term that does not contain any variables.	In the expression x – 2, the constant term is –2.
A coefficient is a number being multiplied by a variable.	In the expression 3m²+ 4m – 1, the coefficients are 3 and 4.
A leading coefficient is the number being multiplied by the variable with the highest power.	In the expression 3m²+ 4m – 1, the leading coefficient is 3.

Practice 1

Are the following algebraic expressions polynomials? If not, provide a reason for why not.

a) $4{y}^{2}-8y-6$

b) $\sqrt{5}b$

c) $\dfrac{5}{b}$

d) $12{m}^{3}-2m+5{m}^\(\frac{1}{3}$

e) $2{x}^{5}-5{x}^{3}-9{x}^{2}+3x+4$

f) $15+5{n}^{-3}$

Practice 2

Fill in the following chart. Identify the type, coefficient(s), leading coefficient, and constant term in each polynomial. The type relates to the number of terms. If the expression has four or more terms, it is called a polynomial.

Polynomial

Type

Coefficient(s)

Leading Coefficient

Constant Term

a) $8{x}^{4}-7{x}^{2}-6x-5$

b) $6b-3{a}^{3}+7$

c) $-5{a}^{4}{b}^{2}$

d) $6-3{n}^{2}+5{n}^{3}-8m$

Polynomials are also often classified by degree. It can be useful to identify the degree of a polynomial because it can give us an idea what the graph of a polynomial function will look like. The degree of a polynomial can be determined by the exponents of the variable(s).

Table 4.1.2
Degree of a Polynomial
The degree of a term is the sum of the exponents of its variables. The degree of a polynomial is the highest degree of all its terms. The degree of a constant term is 0.

Practice 3

Find the degree of the following polynomials.

a) $10y$

b) $4{x}^{3}-7x+5$

c) $-15$

d) $-8{b}^{2}+9b-2$

e) $2m+8{m}^{2}-3{m}^{5}$

f) $6{p}^{3}-4{p}^{2}-5$

Adding and Subtracting Polynomials

To add and subtract polynomials, we need to understand what like terms are. Like terms are terms that have the same variable raised to the same exponent. For example, 3x and 5x are like terms, and 9n² and n² are also like terms, but 5y² and –3y are not considered like terms because one term has y² and the other term has y. Like terms can be added or subtracted together. Terms that are not like terms cannot be added or subtracted together.

Table 4.1.3
To add and subtract polynomials:
Step 1: Identify like terms. Step 2: Rearrange the expression so like terms are together. Step 3: Add or subtract the coefficients of the like terms.

Example 1

Simplify the expression $7{x}^{2}+8x+{x}^{2}-4x$

Table 4.1.4
Steps	Solution
Identify the like terms.	$7{x}^{2}\:\text{and}\:{x}^{2}$ are like terms. $8x\:\text{and}\:-4x$ are like terms.
Rearrange the expression so the like terms are together.	$7{x}^{2}+{x}^{2}+8x-4x$
Add or subtract the coefficients of the like terms.	$8{x}^{2}+4x$

Practice 4

Simplify the expressions.

a) $7x+9+9x+8$

b) $5y+2+8y+4y+5$

c) $3{x}^{2}+9x+{x}^{2}+5x$

d) $10{x}^{3}+9y+10+2{x}^{3}+5y-2x$

When adding or subtracting polynomials that include brackets, you will need to distribute the sign in front of the brackets prior to combining like terms.

Example 2

Simplify the expression $\left(9{w}^{2}-7w+5\right)-\left(2{w}^{2}-4\right)$

Table 4.1.5
Steps	Solution
Distribute and identify like terms.	$9{w}^{2}-7w+5-2{w}^{2}+4$
Rearrange the expression so the like terms are together.	$9{w}^{2}-2{w}^{2}-7w+5+4$
Add or subtract the coefficients of the like terms.	$7{w}^{2}-7w+9$

Practice 5

Simplify the expressions.

a) $\left(7{x}^{2}-4x+5\right)+\left({x}^{2}-7x+3\right)$

b) $\left(8{x}^{2}+3x-19\right)-\left(7{x}^{2}-14\right)$

c) $\left({u}^{2}-6uv+5{v}^{2}\right)+\left(3{u}^{2}+2uv\right)$

d) $\left({p}^{2}+{q}^{2}\right)-\left({p}^{2}+10pq-2{q}^{2}\right)$

Homework

Are the following algebraic expressions polynomials? If not, provide a reason for why not.

a) $\frac{7}{x} + 3x - 5$ b) $4y^{\frac{1}{3}} + 2y^2 - 7$ c) $5x^{-2} + 3x + 1$

d) $3z^2 - 2z + \sqrt{z}$ e) $8a^2 + 5a - 9$ f) $2b + \frac{1}{b} - 4$

g) $6m^3 + 4m^2 + 2m$ h) $5x + 2x^{-\frac{1}{2}} - 4$ i) $7x^3 + 4x^2 - 3x + 2$
Fill in the following chart.

Polynomial Type Coefficient(s) Leading Coefficient Constant Term

a) $4x^3$

b) $5y^2 - 3y$

c) $5x + 2x^2 - 7$

d) $6z^2 + 4z + 1$

e) $7 + 3a^4 - 5a^2$

f) $-b^2 - 3 + 4b + 2b^3$

g) $8m^2 - 2m + 9$

h) $3n^3 + 9n^4 - 4n^2 + 6n$
Find the degree of the following polynomials.

a) $4x^3 + 2x^2 - 5x + 7$ b) $9y^5 - 3y^2 + 4$

c) $2z^4 + 3z^3 - z + 1$ d) $5m^2 - 6m + 3$

e) $7a^6 - 4a^4 + a^3 - 9$ f) $6b^3 + b^2 - 2b + 8$

g) $10x^7 - 5x^5 + 3x^3 - x + 4$ h) $2p^4 + 3p^2 - 7p + 1$

i) $5$ j) $4z^2$
Simplify the expressions.

a) $17a+9a$ b) $4c+2c+c$

c) $9x+3x+8$ d) $7u+2+3u+1$

e) $7p+6+5p+4$

f) $10a+7+5a-2+7a-4$

g) $3{x}^{2}+12x+11+14{x}^{2}+8x+5$

h) $5x^2+3x+2x^2+4x$

i) $5p^2 - 3q + 2p^2 + 7q$

j) $7a^2 + 5a + 3a^2 + 6a$

k) $3x^2 + 2y + 4x^2 - 5y + 6x - 2y + 7x^2$

l) $7a^2 + 4b + 3a - 2b^2 + 5b + a^2 - 4b$

m) $5p^2 - 3q + 2p^2 + 7q - p^2 + 4q + 2p$

n) $6m^2 - 3n + 2m + 4n^2 + 7m - n$

o) $8x^2 - 4y + 5x - 3y^2 + 2x - y + 7x^2$

p) $9a^2 + 3b + 2a - b^2 + 4b + a^2 + b$

q) $4x^{2}+3x-5z+4z^{2}+4x^{2}+4x-5z$

r) $x+2y^{2}+5y-4y^{2}$
Simplify the expressions.

a) $\left( 5{x}^{2}-2x+3 \right) + \left( 3{x}^{2}-4x+7 \right)$

b) $\left( 10{y}^{2}+5y-6 \right) + \left( 4{y}^{2}-3y+8 \right)$

c) $\left( 6{x}^{2}+5x-12 \right) - \left( 2{x}^{2}-6 \right)$

d) $\left( 8{b}^{2}-6b+7 \right) - \left( 4{b}^{2}-2b+3 \right)$

e) $\left( 4{u}^{2}-8uv+7v \right) + \left( 5{u}^{2}+3uv-2v \right)$

f) $\left( 2{p}^{2}+3{q}^{2} \right) - \left( {p}^{2}+8pq-3{q}^{2} \right)$

g) $\left( 4{x}^{2}+2x-3 \right) + \left( 6{x}^{2}+4x-8 \right)$

h) $\left( 5a^{2}-3a+2 \right) + \left( 8a^{2}+2a-6 \right)$

i) $\left( 7p^{2}+3p-4 \right) - \left( 4p^{2}-5p+2 \right)$

j) $\left( 12x^{2}-6x+7 \right) - \left( 3x^{2}+4x-5 \right)$

k) $\left( 6m^{2}-4m+5 \right) + \left( 3m^{2}+5m-2 \right)$

l) $\left( 5q^{2}-2q+9 \right) - \left( 2q^{2}+3q-1 \right)$

Answers

1.

a) No, because it contains a term with a variable in the denominator.	b) No, because it contains a fractional exponent.	c) No, because it contains a negative exponent.
d) No, because it contains a fractional exponent.	e) Yes, it is a polynomial.	f) No, because it contains a term with a variable in the denominator.
f) Yes, it is a polynomial.	h) No, because it contains a negative fractional exponent.	i) Yes, it is a polynomial.

2.

Polynomial	Type	Coefficient(s)	Leading Coefficient	Constant Term
a) $4x^3$	Monomial	$4$	$4$	$0$
b) $5y^2 - 3y$	Binomial	$5$ , $-3$	$5$	$0$
c) $5x + 2x^2 - 7$	Trinomial	$2$ , $5$	$2$	$-7$
d) $6z^2 + 4z + 1$	Trinomial	$6$ , $4$	$6$	$1$
e) $7 + 3a^4 - 5a^2$	Trinomial	$3$ , $-5$	$3$	$7$
f) $-b^2 - 3 + 4b + 2b^3$	Polynomial	$2$ , $-1$ , $4$	$2$	$-3$
g) $8m^2 - 2m + 9$	Trinomial	$8$ , $-2$	$8$	$9$
h) $3n^3 + 9n^4 - 4n^2 + 6n$	Polynomial	$9$ , $3$ , $-4$ , $6$	$9$	$0$

3.

a) 3	b) 5	c) 4	d) 2	e) 6
f) 3	g) 7	h) 4	i) 0	j) 2

4.

a) $26a$	b) $7c$	c) $12x+8$
d) $10u+3$	e) $12p+10$	f) $22a+1$
g) $17{x}^{2}+20x+16$	h) $7x^2+7x$	i) $7p^2 + 4q$
j) $10a^2 + 11a$	k) $14{x}^{2}+6x-5y$	l) $8a^2 - 2b^2 + 3a + 5b$
m) $6p^{2}+2p+8q$	n) $6m^2 + 4n^2 + 9m - 4n$	o) $15x^2 - 3y^2 + 7x - 5y$
p) $10a^2 - b^2 + 2a + 8b$	q) $8x^2 + 4z^2 + 7x - 10z$	r) $-2y^2 + x + 5y$

5.

a) $8{x}^{2}-6x+10$	b) $14{y}^{2}+2y+2$	c) $4{x}^{2}+5x-6$
d) $4{b}^{2}-4b+4$	e) $9{u}^{2}-5uv+5v$	f) ${p}^{2}+6{q}^{2}-8pq$
g) $10{x}^{2}+6x-11$	h) $13a^{2}-a-4$	i) $3p^{2}+8p-6$
j) $9x^{2}-10x+12$	k) $9m^{2}+m+3$	l) $3q^{2}-5q+10$

Attribution

Unless otherwise indicated, material on this page has been adapted from the following resources:

Kuczynska, A. (2019). Intermediate algebra and trigonometry. BCcampus. https://collection.bccampus.ca/textbooks/intermediate-algebra-and-trigonometry-bccampus-165/, licensed under CC BY-NC 4.0

Marecek, L., Anthony-Smith, M., & Honeycutt, M. (2020). Elementary algebra 2e. OpenStax. https://collection.bccampus.ca/textbooks/elementary-algebra-2e-openstax-106/, licensed under CC BY 4.0

Mazur, I. (2021). Introductory algebra. BCcampus. https://collection.bccampus.ca/textbooks/intermediate-algebra-bccampus-412/, licensed under CC BY 4.0

Wang, M. (2018). Key concepts of intermediate level math. BCcampus. https://collection.bccampus.ca/textbooks/key-concepts-of-intermediate-level-math-bccampus-204/, licensed under CC BY 4.0

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