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Chapter 2

Introduction to Algebra

Abstract

This chapter provides an introduction to algebra through learning basic principles of addition, multiplication, exponents, and radicals. The chapter shows addition, subtraction, multiplication, and division of polynomials. Also, the chapter introduces ratios, proportions, and variation.

Keywords

Algebra; Exponent; Radical; Ratio; Polynomial; Proportion; Variation

When solving problem, dig at the roots instead of just hacking at the leaves.

Anthony J. D'Angelo

James Prescott Joule (1784–1858), an English physicist who studied the nature of heat and established its relationship to mechanical work. He therefore laid the foundation for the theory of conservation of energy, which later influenced the First Law of Thermodynamics. He also formulated the Joule's laws which deal with the transfer of energy.

Introduction

In this chapter, we will discuss useful and important basic topics of introductory algebra that are essential for easier understanding of technical and applied mathematics.

2.1. Introduction to algebra

In algebra, some values are known but others are either unknown or not specified, which can be represented by letters or variables. Algebra performs mathematical operations on numbers and variables (unknown numbers), in order to find the values of variables.

Algebra uses signs to represent operations. The study of algebra involves the use of equations to solve various problems. Today, technicians and engineers are required to learn algebra to solve general and specific problems related to their fields and new technologies.

In algebra, an expression is a combination of letters that represent numbers known as variables, separated by operations creating terms. For example, 3x − 1 is an expression with two terms. The number or constant in front of the variable of an expression is called a coefficient. For the above example, 3 is the coefficient of x. An expression is a mathematical statement that does not contain an equal sign (=), but it can contain numbers, variables, and operators (+, –, ×, ÷). Equations are expressions followed by an equal sign.

In algebra, we use formulas that are shortened rules for employing letters as symbols to represent quantities. For example, we use the letter l to represent the length, the letter w to represent the width, and the letter A for area. We can write the formula for the area of a rectangle as A = l × w.

Example 1

Find the coefficients of the following expressions:

(a) 25x − y

(b) 7t + 3z

Solution

(a) 25 is the coefficient of x and −1 is the coefficient of y

(b) 7 is the coefficient of t and 3 is the coefficient of z

Some expressions contains more than one grouping of symbols in parentheses ( ) or brackets [ ]. In this case, you need to compute the innermost grouping first and work outward.

When terms involve the same letter with different coefficients in an expression, we call them like terms. For example, the expression 5x − 2y + 7x + 4y have two like terms 5x with 7x and −2y with 4y. Usually an expression can be simplified by adding and subtracting the like terms (apples to apples and oranges to oranges).

Example 2

Compute the following

(a) x(a + 3a) − 2

(b) 5[x(a + 3a) − 2]

Solution

(a) First we solve the operation in parentheses, which is a + 3a = 4a, then we multiply the result 4a by x, which is 4ax, then we subtract 2 from 4ax to give us 4ax − 2.

(b) First we solve the operation in parentheses from a + 3a = 4a, then we multiply the result, 4a, by x, which is 4ax. Then we subtract 2 from 4ax to give us 4ax − 2. Next, we multiply by 5 to give us 20ax − 10.

Example 3

Simplify the expression 8x − 3y − 2x + 15y.

Solution

First, we collect the like terms for x, which are 8x − 2x = 6x, and the same for the y as −3y + 15y = 12y. Therefore, the expression 8x − 3y −2x + 15y can be simplified to 6x + 12y.

Substitution is used in algebra to find a numerical value of a variable in an expression. It can be done by assigning a particular number value to a variable.

Example 4

Let y = x − 2 and x = 5, find the value of y.

Solution

We need to substitute 5 for x to get the value of y as y = 5 – 2 = 3.

2.1.1. Basic principles of addition

We can summarize the basic principles of addition as in the table below considering a, b, and c are real numbers.

Property	Symbol
Commutative	a + b = b + a
Associative	a + (b + c) = (a + b) + c
Distributive	a(b + c) = ab + ac
Identity is 0	0 + a = a
Inverse (opposite)	a + (−a) = 0

We will apply the above principles on algebra as in the below examples. We use x and y to represent variables.

Example 1

(a) 5x + 2y = 2y + 5x

(b) 7x + (3y + 9) = (7x + 3y) + 9

(d) 0 + 6x = 6x

(e) 7x + (−7x) = 0

2.1.2. Basic principles of multiplication

The basic principles of multiplication are summarized as in the below table with consideration that a, b, and c are real numbers.

Properties of multiplication

Property	Symbol
Commutative	ab = ba
Associative	a(bc) = (ab)c
Distributive	(b + c)a + ba + ca
Identity is 1	a·1 = a
Inverse (reciprocal)	$a (\frac{1}{a}) = 1$ $a (\frac{1}{a}) = 1$ , a ≠ 0

Again, we applied these principles to algebra as in the below examples. It is important to know that ab = a × b = a·b.

Example 1

(a) (5x)(2y) = (2y)(5x) = 10xy

(b) 7x [(3y)(9)] = [(7x)(3y)]9 = 189xy

(d) 1·6x = 6x

(e) 7x (1/7x) = 1

(f) x·6 = 6x

(g) (–9t)t = −9t²

We summarized some of the most basic algebraic operations and the way they simplified as in the below table with consideration that a, b, c, and d are real numbers.

Operation	Simplified to
a + (−b)	a − b
$a \times \frac{1}{b}$ $a \times \frac{1}{b}$ , b ≠ 0	$\frac{a}{b}$ $\frac{a}{b}$
$\frac{a}{a}$ $\frac{a}{a}$ , a ≠ 0	1
a(−b)	−(ab)
(−a)b	−(ab)
(−a)(−b)	ab
−(−a)	a
$\frac{a}{- b}$ $\frac{a}{- b}$	$\frac{- a}{b}$ $\frac{- a}{b}$ or $- \frac{a}{b}$ $- \frac{a}{b}$
$\frac{- a}{- b}$ $\frac{- a}{- b}$	$\frac{a}{b}$ $\frac{a}{b}$
ac = bc, c ≠ 0	a = b
$\frac{a c}{b c}$ $\frac{a c}{b c}$ , b ≠ 0 and c ≠ 0	$\frac{a}{b}$ $\frac{a}{b}$
If ab = 0	a = 0 or b = 0, or a = b = 0
$\frac{a}{b} + \frac{c}{d}$ $\frac{a}{b} + \frac{c}{d}$ , b ≠ 0 and d ≠ 0	$\frac{a d + b c}{b d}$ $\frac{a d + b c}{b d}$
$\frac{a}{b} \times \frac{c}{d}$ $\frac{a}{b} \times \frac{c}{d}$	$\frac{a c}{b d}$ $\frac{a c}{b d}$
$\frac{(\frac{a}{b})}{(\frac{c}{d})}$ $\frac{(\frac{a}{b})}{(\frac{c}{d})}$ , b ≠ 0, d ≠ 0, and c ≠ 0	$\frac{a}{b} \times \frac{d}{c} = \frac{a d}{b c}$ $\frac{a}{b} \times \frac{d}{c} = \frac{a d}{b c}$
a − (b + c)	a – b − c
a − (b − c)	a – b + c

When several operations exist in a single expression, then we need to follow the order of operations as we start first with parentheses, second with exponent, third with multiplication or division, left to right, and fourth with addition or subtraction, left to right.

Example 2

Simplify the following:

(a) 55 − (6 − 2) × 3²

(b) 7[5 −2(a + b)]

(d)

x \times \frac{1}{y}

$x \times \frac{1}{y}$

(e) x × 0

(f) 0/x

(g)

\frac{y}{y}

$\frac{y}{y}$

(h) x(−y)

(i) (−x)y

(j) (−x)(−y)

(k) −(−x)

(l)

\frac{x}{- y}

$\frac{x}{- y}$

(m)

\frac{- x}{- y}

$\frac{- x}{- y}$

(n) 3x = 3y

(o)

\frac{7 x}{7 y}

$\frac{7 x}{7 y}$

(p) a(b = 0)

(q)

\frac{x}{3} + \frac{y}{4}

$\frac{x}{3} + \frac{y}{4}$

(r)

\frac{7}{x} \times \frac{2}{y}

$\frac{7}{x} \times \frac{2}{y}$

(s)

\frac{(\frac{2}{x})}{(\frac{9}{y})}

$\frac{(\frac{2}{x})}{(\frac{9}{y})}$

(t) x − (3z + t)

(u) 7z − (y − x)

Solution

(a) 55 − (6 − 2) × 3² = 55 – 4 × 3², we start first with parentheses

=55 – 4 × 9, then exponent

=55 − 36, multiplication

=19, last subtraction

(b) 7[5 − (a + b)] = 7[5 – a − b] = 35 − 7a − 7b

(d)

x \times \frac{1}{y} = \frac{x}{y}

$x \times \frac{1}{y} = \frac{x}{y}$

(e) x × 0 = 0

(f)

\frac{0}{x} = 0

$\frac{0}{x} = 0$

(g)

\frac{y}{y} = 1

$\frac{y}{y} = 1$

(h) x(−y) = −(xy) = −xy

(i) (−x)y = −(xy) = −xy

(j) (−x)(−y) = xy

(k) −(−x) = x

(l)

\frac{x}{- y} = \frac{- x}{y} = - \frac{x}{y}

$\frac{x}{- y} = \frac{- x}{y} = - \frac{x}{y}$

(m)

(n)

(o)

(p) a(b = 0) = a × 0 = 0

(q)

\frac{x}{3} + \frac{y}{4} = \frac{4 \times x + 3 \times y}{3 \times 4} = \frac{4 x + 3 y}{12}

$\frac{x}{3} + \frac{y}{4} = \frac{4 \times x + 3 \times y}{3 \times 4} = \frac{4 x + 3 y}{12}$

(r)

\frac{7}{x} \times \frac{2}{y} = \frac{7 \times 2}{x \times y} = \frac{14}{x y}

$\frac{7}{x} \times \frac{2}{y} = \frac{7 \times 2}{x \times y} = \frac{14}{x y}$

(s)

\frac{(\frac{2}{x})}{(\frac{9}{y})} = \frac{2}{x} \times \frac{y}{9} = \frac{2 y}{9 x}

$\frac{(\frac{2}{x})}{(\frac{9}{y})} = \frac{2}{x} \times \frac{y}{9} = \frac{2 y}{9 x}$

(t) x − (3z + t) = x − 3z − t

(u) 7z − (y − x) = 7z – y + x

Example 3

Simplify the following:

(a)

\frac{- 2}{3} + \frac{5}{2}

$\frac{- 2}{3} + \frac{5}{2}$

(b)

\frac{- 9 x^{2}}{- 3 x}

$\frac{- 9 x^{2}}{- 3 x}$

(c)

\frac{5}{3} \cdot \frac{11}{2}

$\frac{5}{3} \cdot \frac{11}{2}$

(d)

\frac{12 / 5}{3 / 7}

$\frac{12 / 5}{3 / 7}$

(e) 3x (2 − 7x)

(f) 6x – 2 – 3x)

(g) 5y − (−5x + 2y)

(h) (−7)(3x)

(i) (−2x)(−3y)

(j) (3x − 2)(3x + 2)

(k)

5 \times \frac{1}{6} + 3

$5 \times \frac{1}{6} + 3$

Solution

(a)

\frac{- 2}{3} + \frac{5}{2} = \frac{- 4 + 15}{6} = \frac{11}{6}

$\frac{- 2}{3} + \frac{5}{2} = \frac{- 4 + 15}{6} = \frac{11}{6}$

(b)

\frac{- 9 x^{2}}{- 3 x} = 3 x

$\frac{- 9 x^{2}}{- 3 x} = 3 x$

(c)

\frac{5}{3} \cdot \frac{11}{2} = \frac{55}{6}

$\frac{5}{3} \cdot \frac{11}{2} = \frac{55}{6}$

(d)

\frac{12 / 5}{3 / 7} = \frac{12}{5} \cdot \frac{7}{3} = \frac{28}{5}

$\frac{12 / 5}{3 / 7} = \frac{12}{5} \cdot \frac{7}{3} = \frac{28}{5}$

(e) 3x (2 − 7x) = 6x − 21x²

(f) 6x – (2 – 3x) = 9x – 2

(g) 5y − (−5x + 2y) = 3y + 5x

(h) (−7)(3x) = −21x

(i) (−2x)(−3y) = 6xy

(j) (3x − 2)(3x + 2) = 9x² − 4

(k)

5 \times \frac{1}{6} + 3 = \frac{5 + 18}{6} = \frac{23}{6}

$5 \times \frac{1}{6} + 3 = \frac{5 + 18}{6} = \frac{23}{6}$

We summarized some of the most common expansions and factors and the way they simplified as in the below table with consideration that a and b are variables.

Expansion and factors

1. a² − b² = (a − b)(a + b)

2. a³ − b³ = (a − b)(a² + ab + b²)

3. aⁿ − bⁿ = (a − b)(aⁿ⁻¹ + aⁿ⁻²b +….+ bⁿ⁻¹), for n an odd integer

4. aⁿ + bⁿ = (a + b)(a^n–1 − aⁿ⁻²b +….− bⁿ⁻¹), for n an even integer

5. a³ + b³ = (a + b)(a² – ab + b²)

6. (a ± b)² = (a ± b)(a ± b) = a² ± 2ab + b²

7. (a ± b)³ = a³ ± 3a²b + 3ab² ± b³

Example 4

(a) (x − y)(x + y) = x² − y²

(b) (x − z)(x² + xz + z²) = x³ − z³

(d) (x + y)² = (x + y)(x + y)

2.1.3. Exponent and radicals

Definition of exponent

If n is a natural number, then

a^{n} = a \cdot a \cdot a \dots \cdot a

$a^{n} = a \cdot a \cdot a \dots \cdot a$

, where a appears as a factor n times.

The expression aⁿ is a combination of the power n, which is the exponent, and a which is the base.

So, a power or exponent indicates how many times the base is multiplied by itself. The power is written in exponent form to represent the number of factors.

For example, 5 × 5 is called the second power and can be written in exponent form as 5².

We read 5² as 5 power of 2. The 5 is called the base and the 2 is called the exponent. When no exponent is written with a number, it is assumed to be one. For example, 5 = 5¹.

Example 1

Find the value of 3⁴.

Solution

3⁴ = 3 × 3 × 3 × 3 = 81.

Based on the definition of exponent, we can define the exponent aⁿ when n is zero or negative values.

Definition of zero and negative exponents

If a is a nonzero real number and n is a positive integer, then

a⁰ = 1 and

a^{- n} = \frac{1}{a^{n}}

$a^{- n} = \frac{1}{a^{n}}$

We can summarize the rules of power as:

Properties of exponents

For any integers n and m, and any real numbers a and b, the following properties are true:

1. aⁿ × a^m = a^n+m

2. (aⁿ)^m = a^n×m

\frac{a^{n}}{a^{m}} = a^{n - m}

$\frac{a^{n}}{a^{m}} = a^{n - m}$

4. (ab)ⁿ = aⁿ × bⁿ

{(\frac{a}{b})}^{n} = \frac{a^{n}}{b^{n}}

${(\frac{a}{b})}^{n} = \frac{a^{n}}{b^{n}}$

We summarize the operations with zero as in the below table with consideration that a is a real number.

Zero properties

1. a – a = 0

2. a × 0 = 0

\frac{0}{a} = 0

$\frac{0}{a} = 0$

, if a ≠ 0

4. a⁰ = 1, if a ≠ 0

5. 0^a = 0

\frac{a}{0}

$\frac{a}{0}$

= undefined

7. 0⁰ = undefined

Definition of radicals

If n is an even or odd natural number and a > 0, then

\sqrt[n]{a} = a^{\frac{1}{n}}

$\sqrt[n]{a} = a^{\frac{1}{n}}$

For a and b are real numbers, and m and n are natural numbers.

We summarize the common properties of square root as in the below table.

Square root properties

$\sqrt[n]{a}$ $\sqrt[n]{a}$ , a: real number, n: integer number	$a^{\frac{1}{n}}$ $a^{\frac{1}{n}}$
$\sqrt[n]{a^{m}}$ $\sqrt[n]{a^{m}}$ a: real number, n and m: integer numbers	$a^{\frac{m}{n}}$ $a^{\frac{m}{n}}$
$\sqrt[n]{a}$ $\sqrt[n]{a}$ $\sqrt[n]{b}$ $\sqrt[n]{b}$	$\sqrt[n]{a b}$ $\sqrt[n]{a b}$
$\frac{\sqrt[n]{a}}{\sqrt[n]{b}}$ $\frac{\sqrt[n]{a}}{\sqrt[n]{b}}$ , b ≠ 0	$\sqrt[n]{\frac{a}{b}}$ $\sqrt[n]{\frac{a}{b}}$
$\sqrt[m]{\sqrt[n]{a}}$ $\sqrt[m]{\sqrt[n]{a}}$	$\sqrt[m n]{a}$ $\sqrt[m n]{a}$

Example 2

(a)

\sqrt[2]{x} = \sqrt{x} = x^{\frac{1}{2}}

$\sqrt[2]{x} = \sqrt{x} = x^{\frac{1}{2}}$

(b)

\sqrt[6]{y} = y^{\frac{1}{6}}

$\sqrt[6]{y} = y^{\frac{1}{6}}$

(c)

\sqrt[7]{x^{3}} = x^{\frac{3}{7}}

$\sqrt[7]{x^{3}} = x^{\frac{3}{7}}$

(d)

\sqrt[5]{x} \sqrt[5]{y} = \sqrt[5]{x y}

$\sqrt[5]{x} \sqrt[5]{y} = \sqrt[5]{x y}$

(e)

\frac{\sqrt[3]{x}}{\sqrt[3]{y}} = \sqrt[3]{\frac{x}{y}}

$\frac{\sqrt[3]{x}}{\sqrt[3]{y}} = \sqrt[3]{\frac{x}{y}}$

(f)

\sqrt[5]{\sqrt[3]{x}} = \sqrt[15]{x}

$\sqrt[5]{\sqrt[3]{x}} = \sqrt[15]{x}$

2.1. Exercises

In problems 1–6, perform the indicated operations using exponents, and write each answer in a simple form.

1. 3 × 3

2. 5 × 5 × 5

3. 2 × 16

4. 3 × 9

5. mn × mn × mn × mn

6. vt × vt × vt

In problems 7–22, find the value for n for which each statement is true.

7. 2⁴ × 2⁵ = 2ⁿ

8. 2⁴ × 2⁶ × 2⁸ = 4ⁿ

9. 3ⁿ × 3⁵ = 3¹¹

10. 3ⁿ × 3⁹ × 3⁴ = 3²¹

11. 7(7² × 7³) = 7ⁿ

12. 5⁸(5⁴ × 5) = 5ⁿ

13. 2ⁿ × 2ⁿ = 1

14. 2ⁿ × 2³ⁿ = 1

15. 3^n–1 = 3

16. 3²ⁿ⁺¹ = 3⁴

17. 2⁵ × 3⁵ = 6ⁿ

18. 2⁷ × 3⁷ = 6ⁿ⁻¹

19. 2¹⁰ × 5⁷ = 2³ × 10ⁿ⁺¹

20. 3¹² × 5⁸ = 3⁴ × 15ⁿ⁻¹

21. 4⁵ × 3⁵ = n⁵

22. 2⁷ × 9⁷ = n⁷

In problems 23–28, perform each of the indicated operations.

23. 2⁷

24. 2⁸

25. (2³)²

26. (3²)³

27. (2·3²)²

28. (5³ × 3²)⁰

In problems 29–36, write each rational expression in lowest terms.

29.

\frac{3 t^{2}}{9 t}

$\frac{3 t^{2}}{9 t}$

30.

\frac{21 r^{3}}{7 r}

$\frac{21 r^{3}}{7 r}$

31.

\frac{2 z - 12}{6 z - 36}

$\frac{2 z - 12}{6 z - 36}$

32.

\frac{5 p + 15}{7 p + 21}

$\frac{5 p + 15}{7 p + 21}$

33.

\frac{5 (p + 11)}{(p + 11) (p - 5)}

$\frac{5 (p + 11)}{(p + 11) (p - 5)}$

34.

\frac{2 (p + 4)}{(p + 4) (2 p - 4)}

$\frac{2 (p + 4)}{(p + 4) (2 p - 4)}$

35.

\frac{9 m^{2} + 18 m}{3 m}

$\frac{9 m^{2} + 18 m}{3 m}$

36.

\frac{4 m^{2} + 32 m}{2 m}

$\frac{4 m^{2} + 32 m}{2 m}$

2.2. Addition, subtraction, multiplication, and division of monomials

A monomial is one variable that is the product of a constant (a letter that stands just for one number) and a variable with a nonnegative integer power. Recall, integer is a set of {…, −2, −1, 0, 1, 2,…}

A monomial form = cx^k, k ≥ 0

c = a constant (coefficient of the monomial)

x = a variable

k = an integer (degree of monomial)

Exponent (power) is a small number above the variable or number (called base) to indicate the number of times a number is multiplied by itself.

For

x^{k} = x \cdot x \cdot x \dots x

$x^{k} = x \cdot x \cdot x \dots x$

k factors

x = base

k = exponent

Example 1

(a) x⁰ = 1

(b) x¹ = x

(d)

x^{- 9} = \frac{1}{x^{9}}

$x^{- 9} = \frac{1}{x^{9}}$

(e)

x^{4} = \frac{1}{x^{- 4}}

$x^{4} = \frac{1}{x^{- 4}}$

Example 2

7x³ is a monomial with a coefficient 7 and of a degree 3.

Note that

x^{- k} = \frac{1}{x^{k}}

$x^{- k} = \frac{1}{x^{k}}$

, if k ≠ 0

Two monomials can use addition and subtraction between them only when they are like terms (same variable with same degree).

Example 3

(a) 3x² + 5x² = (3 + 5)x² = 8x²

(b) 3x² + 5x ≠ 8x²

(d) 5I³ − 13I³ = (5 − 13)I³ = −8I³

A term is an expression, for example, 6t⁵, is a term with number 6 as coefficient and t as variable, and 5 as the exponent. The expression t⁵ means t·t·t·t·t. There are two types of terms: first, like terms, which are terms having the same variable and same exponent; second, unlike terms, which are terms that do not have the same variable and the same exponent. For example, z and z² are unlike terms, whereas, 8z⁵ and 3z⁵ are like terms.

2.2.1. Rules for multiplication of monomials

The rules for multiplication of monomials are summarized in the below table.

c₁a^m·c₂aⁿ = c₁c₂a^m+n, m, n are integers

(a^m)ⁿ = a^mn

(ca)ⁿ = cⁿaⁿ

Example 1

(a) 3x²·2x⁵ = (3·2)x²⁺⁵ = 6x⁷

(b) (x²)⁵ = x^2·5 = x¹⁰

2.2.2. Rules for division of monomials

The rules for multiplication of monomials are summarized in the below table.

(\frac{a^{n}}{a^{m}}) = a^{n - m}

$(\frac{a^{n}}{a^{m}}) = a^{n - m}$

, m, n are integers

{(\frac{a}{c})}^{m} = (\frac{a^{m}}{c^{m}})

${(\frac{a}{c})}^{m} = (\frac{a^{m}}{c^{m}})$

\frac{1}{a^{n}} = a^{- n}

$\frac{1}{a^{n}} = a^{- n}$

Example 1

Simplify the following

(a)

(\frac{x^{5}}{x^{2}})

$(\frac{x^{5}}{x^{2}})$

(b)

{(\frac{x}{4})}^{2}

${(\frac{x}{4})}^{2}$

(c)

\frac{1}{x^{7}}

$\frac{1}{x^{7}}$

(d)

\frac{5}{x^{4}}

$\frac{5}{x^{4}}$

(e) 19x⁻³

Solution

(a)

(\frac{x^{5}}{x^{2}}) = x^{5} \cdot x^{- 2} = x^{3}

$(\frac{x^{5}}{x^{2}}) = x^{5} \cdot x^{- 2} = x^{3}$

or by cancellation icon

(b)

{(\frac{x}{4})}^{2} = (\frac{x^{2}}{4^{2}}) = \frac{x^{2}}{16}

${(\frac{x}{4})}^{2} = (\frac{x^{2}}{4^{2}}) = \frac{x^{2}}{16}$

(c)

\frac{1}{x^{7}} = x^{- 7}

$\frac{1}{x^{7}} = x^{- 7}$

(d)

\frac{5}{x^{4}} = 5 x^{- 4}

$\frac{5}{x^{4}} = 5 x^{- 4}$

(e)

19 x^{- 3} = \frac{19}{x^{3}}

$19 x^{- 3} = \frac{19}{x^{3}}$

A binomial is the sum or difference of two monomials that are not like terms.

For example, x + x² is a binomial with two terms.

2.2.3. Addition, subtraction, multiplication, and division of polynomials

A polynomial is a monomial or a finite sum of monomials. In other words, a polynomial is a term or a finite sum of terms in which all variables have exponents of whole numbers and no variables with negative exponents.

For example, the following are polynomials:

−5, 3n, and 6y³ + 2y² + x.

The general form of a polynomial is

a_nxⁿ + a_n−1x^n–1 +…+ a₁x + a₀

Where

a_n, a_n−1,…a₁x, a₀ = constants (coefficients of the polynomial)

n: an integer (n ≥ 0) and is the degree of the polynomial

x: a variable.

For example, 5z³ + 3z − 1 is polynomial with coefficients 5, 3, −1 with degree of 3. Note that

\frac{1}{t^{2}}

$\frac{1}{t^{2}}$

is not a polynomial.

A polynomial can have more than one variable. A term (monomials that make up a polynomial) with more than one variable has a degree of the sum of all the exponents of the variables.

The degree of a polynomial in more than one variable is equal to the greatest degree of any term in the polynomial.

For example, 5z³x⁶ has a degree of 9, and z³x + 3xy − 1 has a degree of 4.

There are four types of polynomials:

1. Trinomial: a polynomial containing exactly three terms

2. Binomial: a polynomial containing exactly two terms

3. Monomial: a single term polynomial

4. Just polynomial

Example 1

1. z³x + 3xy − 1 is polynomial of type trinomial

2. z³x + 3x is polynomial of type binomial

3. 3xy is polynomial of type monomial

4. z³x + 3xy + 7t − 17n is polynomial

Polynomials are added and subtracted by combining the like terms.

Example 2

Add or subtract the following polynomials:

(a) (2z³ + 4z³ − 3z) − (z³ + 6z² − z) + 17

(b) −3(z² − 4z −3) + (z² + z)

Solution

(a) Distribution of the minus sign and then combining like terms.

(2z³ + 4z³ − 3z) − (z³ + 6z² − z) + 17

=2z³ + 4z³ − 3z − z³ − 6z² + z + 17

=(2z³ + 4z³ − z³) − 6z² + (−3z + z) + 17

=5z³ − 6z² − 2z + 17.

(b) Distribution of the negative 3 and then combining like terms.

−3(z² − 4z − 3) + (z² + z)

=−3z² + 12z + 9 + z² + z

=(−3z² + z²) + (12z + z) + 9

=−2z² + 13z + 9.

Products of polynomials are found by using the associative, distributive, and exponent properties.

Example 3

Multiply the polynomials

(a) 5V(2V − 3)

(b) (t + 3)(2t² – t + 1)

Solution

(a) 5V(2V − 3) = 5V(2V) − 5V(3) = 10V² − 15V

(b) (t + 3)(2t² − t + 1) = t(2t² – t + 1) + 3(2t² – t + 1) = (2t³ − t² + 1) + (6t² − 3t + 3)

=2t³ + 5t² − 2t + 3

=(3I − 5)[(I + 2)(I + 1)]

=(3I − 5)[I² + 3I + 2]

=3I[I² + 3I + 2] − 5[I² + 3I + 2]

=3I³ + 9I² + 6I − 5I² − 15I − 10

=3I³ + 4I² − 9I − 10

The division of two polynomials can be found using long division, similar to the long division of whole numbers.

Example 4

Divide 2x³ + 3x² + x + 6 by x² − 1

Solution

$\frac{2 x^{3} + 3 x^{2} + x + 6}{x^{2} - 1} = (2 x + 3) + \frac{3 x + 9}{x^{2} - 1}$ $\frac{2 x^{3} + 3 x^{2} + x + 6}{x^{2} - 1} = (2 x + 3) + \frac{3 x + 9}{x^{2} - 1}$

Note that, (Quotient) (Divisor) + Remainder = Dividend.

2.2. Exercises

Perform the indicated algebraic operations.

1. (−V² − 5V + 1) + (2V² + 6V − 1)

2. (9V³ + 3V² − 7V) − (2V³ − 6V² – V − 3)

3. 3(2I² − 5I + 7) + 2(−I² + 2I − 3)

4. −4(I² + 5I − 1) − 2(−I² − 3I + 6)

5. −2t(3t³ + 4t² − 5t − 6)

6. 3t(2t^1/2 − 4t² + 3t − 5)

7. (5q + 2)(3q + 4)

8. (q − 1)(2q + 3)

9. (2r + t)²

10. (r − 2t)²

11. (2k + 1)(3k² + 4k − 1)

12. (k − 1)(2k² − 3k + 8)

13. (p + m + n)(3p − 4m − 2n)

14. (p + 2m − n)(p − 3m − 6n)

15. (p + 1)(2p + 3)(p − 2)

16. (4p + 5)(3p − 2)(2p − 1)

17.

\frac{z^{3} + 2 z^{2} + z + 6}{z + 1}

$\frac{z^{3} + 2 z^{2} + z + 6}{z + 1}$

18.

\frac{5 z^{3} - 2 z^{2} - z + 9}{z^{2} + 1}

$\frac{5 z^{3} - 2 z^{2} - z + 9}{z^{2} + 1}$

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Table of Contents for Chapter 2. Introduction to Algebra

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Abstract

Keywords

Introduction

2.1. Introduction to algebra

Example 1

Solution

Example 2

Solution

Example 3

Solution

Example 4

Solution

2.1.1. Basic principles of addition

Example 1

2.1.2. Basic principles of multiplication

Example 1

Example 2

Solution

Example 3

Solution

Example 4

2.1.3. Exponent and radicals

Example 1

Solution

Example 2

2.1. Exercises

2.2. Addition, subtraction, multiplication, and division of monomials

Example 1

Example 2

Example 3

2.2.1. Rules for multiplication of monomials

Example 1

2.2.2. Rules for division of monomials

Example 1

Solution

2.2.3. Addition, subtraction, multiplication, and division of polynomials

Example 1

Example 2

Solution

Example 3

Solution

Example 4

Solution

2.2. Exercises

Table of Contents for
Chapter 2. Introduction to Algebra