Slide: 1

2.1 ALGEBRAIC EXPRESSIONS

Slide: 2

Algebra is a branch of mathematics that is used to analyze and solve day-to-day business and finance problems.

It deals with different relations and operations by using letters and symbols to represent numbers values, etc.

Section 2.1

13 + 1156 − 26

12 × 9

25 ÷ 5

13x + 1156 − 26y

12(9a + 4)

25b ÷ 5

Arithmetic Expressions

Algebraic Expressions

Letters may be used to represent variables or

constants.

Algebra

Slide: 3

dd, mm, and yyyy are constants. Their values stay the

same, since Tracy`s birthday is on a

fixed date.

When a certain letter of the alphabet is used to represent a varying quantity, it is called a variable. When a letter is used to represent a specific quantity that does not change,

it is called a constant.

Section 2.1

Example Tracy was born on July 20, 2006. She is 5 years old.

Tracy’s agex years

Tracy’s birthdaydd/mm/yyyy

Here, x is a variable. It changes every year as

Tracy gets older.

Variables and Constants

Slide: 4

Temperature in the shade.

Section 2.1

Example Let’s assume that it is ten degrees hotter in the sun than it is in the shade.

y = x + 10

Temperature in the sun.

The temperature may vary, so x and y are free to change, but

the relationship between them is fixed.

When a certain letter of the alphabet is used to represent a varying quantity, it is called a variable. When a letter is used to represent a specific quantity that does not change,

it is called a constant.

Variables and Constants

Slide: 5

A term is a number, variable, or a combination of numbers and variables which are multiplied and/or divided together. Terms are separated by addition and

subtraction operators. A combination of terms makes up an expression.

Section 2.1

4x2 + − 25yx

1st Term 2nd Term 3rd Term

Coefficient:the numerical factor in front of the variable in a term.

Variable

Constant term:a term that has only a number without any variables.

Expression

Algebraic Expressions

Slide: 6

Like terms are terms that have the same variables and exponents.Unlike terms are terms that have different variables or the same variables with different exponents.

Section 2.1

Like terms Unlike terms

Like Terms and Unlike Terms

Slide: 7

Section 2.1

The distinction is important because like terms can be added or subtracted from each other, while unlike terms cannot.

+ =

+ =

+

Like terms are terms that have the same variables and exponents.Unlike terms are terms that have different variables or the same variables with different exponents.

x x 2x

y y 2y

+=x y x y

Like Terms and Unlike Terms

Slide: 8

Factors refer to each of the combinations of variables and/or numbers multiplied together in a term.

Section 2.1

9 and x are factors of the term 9x.

7, x, and y are factors of the term 7xy.

Factors

Slide: 9

Monomial is an algebraic expression that has only one term.Binomial is an algebraic expression with exactly 2 terms.Trinomial is an algebraic expression with exactly 3 terms.

Polynomial is an algebraic expression that has 2 or more terms.

Section 2.1

3

Monomial Binomial Trinomial

Polynomial

5x

9xy

3 + x

5x − 2y

9xy + 21x

3 + x + 4y

5x − 2y + 17xy

9x + 21y − 6z

9x + 21y − 6z − 12

9x + 21y − 6z + 3xyz − 2, and so on…

Number of Terms in an Expression

Slide: 10

Section 2.1

All arithmetic operations can be applied to algebraic expressions.

Addition and Subtraction

When adding or subtracting algebraic expressions, first collect the like terms and group them, then add or subtract the coefficients of the like terms.

Example Add (3x2 + 7x − 1) and (11x2 − 4x +13).

Basic Arithmetic Operations with Algebraic Expressions

Slide: 11

Solution

Subtract (6x2 − 17x + 5) from (14x2 + 3x − 3).

Slide: 12

Section 2.1

All arithmetic operations can be applied to algebraic expressions.

Multiplication: Multiplying Monomials by Monomials.

When multiplying a monomial by a monomial, multiply the coefficients and multiply all variables. If there are any identical variables, use the exponent notation.

Example Multiply 11x2y and 2xy.

Basic Arithmetic Operations with Algebraic Expressions

Slide: 13

Solution

Multiply 9a3, 2ab, and 5b2.

Slide: 14

Section 2.1

All arithmetic operations can be applied to algebraic expressions.

Multiplication: Multiplying Polynomials by Monomials.

When multiplying a polynomial by a monomial, multiply each term of the polynomial by the monomial.

Example Multiply 4x2 and (2x2 + 5x + 3).

Basic Arithmetic Operations with Algebraic Expressions

Slide: 15

Solution

Expand and simplify 12x(x + 3) + x(x − 1).

Slide: 16

Section 2.1

All arithmetic operations can be applied to algebraic expressions.

Multiplication: Multiplying Polynomials by Polynomials.

When multiplying a polynomial by a polynomial, each term of one polynomial is multiplied by each term of the other polynomial.

Example Multiply (2x2 + 3) and (5x + 4).

Basic Arithmetic Operations with Algebraic Expressions

Slide: 17

Solution

Multiply (2x2 + 3) and (x2 + 4x + 1).

Slide: 18

Section 2.1

All arithmetic operations can be applied to algebraic expressions.

Division: Dividing Monomials by Monomials.

When dividing a monomial by a monomial, group the constants and each of the variables separately and simplify them.

Example Divide 12x2y by 9x.

Basic Arithmetic Operations with Algebraic Expressions

Slide: 19

Section 2.1

All arithmetic operations can be applied to algebraic expressions.

Division: Dividing Polynomials by Monomials.

When dividing a Polynomial by a monomial, divide each term of the polynomial by the monomial.

Example Divide (25x3 + 15x2) by 10x.

Basic Arithmetic Operations with Algebraic Expressions

Slide: 20

Solution

Divide (6x3 + 9x4 + 11x) by 9x4.

Slide: 21

Section 2.1

To evaluate algebraic expressions, we replace all the variables with numbers and simplify the expression.

Example Evaluate 5x + 2y, where x = 3 and y = 5.

Evaluating Algebraic Expressions

Slide: 22

Solution

Evaluate the expression , where x = 3, and y = 2.

7xy + 5x

2y + 6

Slide: 23

Solution

Evaluate the expression , where x = 4, and y = 6.20

(2x)2 + 6y

Slide: 24

Section 2.1

Factoring algebraic expressions means finding the common factors for both coefficients and the variables in all the terms.

Example Factor 27x2 + 12x.

Factoring Algebraic Expressions with Common Factors

Slide: 25

Solution

Factor 32xy3 + 8xy4 − 16x3y5 − 2xy3.