Algebraic Expressions
Algebraic Expressions
Slide: 1
Section 2.1
Algebraic Expressions
Slide: 2
Section 2.1
Algebraic Expressions 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.
Arithmetic Expressions
13 + 11 56 − 26 12 × 9 25 ÷ 5
13x + 11 56 − 26y 12(9a + 4) 25b ÷ 5
Algebraic Expressions
Letters may be used to represent variables or constants. Slide: 3
Section 2.1
Writing Simple Algebraic Equations An algebraic equation is a mathematical statement that shows the equality between two algebraic expressions or between a number and an algebraic expression.
Eight more than a number, ‘n’, is 19 28 decreased by twice a number is 10
28 – 2x = 10
The product of 7 and the sum of a number and 9 is 77
7(x + 9) = 77
A number divided by 3 is 8
Slide: 4
n + 8 = 19
n =8 3
Variables and Constants 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.
Example
Let’s assume that it is ten degrees hotter in the sun than it is in the shade.
Temperature in the sun.
Temperature in the shade.
y = x + 10 The temperature may vary, so x and y are free to change, but the relationship between them is fixed. Slide: 5
Algebraic Expressions 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. Expression 1st Term
2nd Term
3rd Term
y 2 4x + x − 25 Variable
Slide: 6
Coefficient:
Constant term:
the numerical factor in front of the variable in a term.
a term that has only a number without any variables.
Algebraic Expressions 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. Expression 1st Term
2nd Term
3rd Term
y 2 4x + x − 25 Variable
Slide: 7
Coefficient:
Constant term:
the numerical factor in front of the variable in a term.
a term that has only a number without any variables.
Like Terms and Unlike Terms 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.
Like terms
2x and 43x 21y2, −6y2, and 15y2 6xy and –2yx Slide: 8
Unlike terms
4y and 8y2 x2, x, and 1
Like Terms and Unlike Terms 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.
The distinction is important because like terms can be added or subtracted from each other, while unlike terms cannot.
=
+ x
x
2x
+
=
y
y
+ x Slide: 9
2y
+
= y
x
y
Number of Terms in an Expression 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.
Monomial
Binomial
Trinomial
3
3+x
3 + x + 4y
5x
5x − 2y
5x − 2y + 17xy
9xy
9xy + 21x
9x + 21y − 6z Polynomial
9x + 21y − 6z − 12 Slide: 10
9x + 21y − 6z + 3xyz − 2, and so on…
Evaluating Algebraic Expressions To evaluate algebraic expressions, we replace all the variables with numbers and simplify the expression.
Example
Slide: 11
Evaluate 5x + 2y, where x = 3 and y = 5.
5 x + 2y = 5 (3) + 2 (5) = 15 + 10 = 25
Evaluate the expression and y = 2. a. b. c. d.
Slide: 12
4.33 5.7 4.75 5.2
7xy + 5x 2y + 6
, where x = 3,
Evaluate the expression and y = 2.
Slide: 13
7xy + 5x 2y + 6
, where x = 3,
Basic Arithmetic Operations with Algebraic Expressions 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).
(3x2 + 7x − 1) + (11x2 − 4x + 13) Identify like terms and group them.
= 3x2 + 7x − 1 + 11x2 − 4x + 13 = 3x2 + 11x2 + 7x − 4x + 13 − 1 = 14x2 + 3x + 12
Slide: 14
Subtract (6x2 − 17x + 5) from (14x2 + 3x − 3).
a. b. c. d.
Slide: 15
−8x2 − 20x + 8 20x2 − 14x + 2 8x2 + 20x − 8 8x2 − 14x − 8
Subtract (6x2 − 17x + 5) from (14x2 + 3x − 3).
Slide: 16
Evaluate the following expression: a2 – {–3a + [7 – (5ab + 9a) – 2a2] – 1}
Slide: 17
Basic Arithmetic Operations with Algebraic Expressions 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.
Group coefficients and variables
11x2y × 2xy = (11)(2)(x2)(x)(y)(y) = 22x3y2
Slide: 18
Multiply 9a3, 2ab, and 5b2.
a. b. c. d.
Slide: 19
90a4b3 90a3b2 16a4b3 16a3b2
Multiply 9a3, 2ab, and 5b2.
Slide: 20
Basic Arithmetic Operations with Algebraic Expressions 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
Slide: 21
Multiply 4x2 and (2x2 + 5x + 3).
Expand and simplify 12x(x + 3) + x(x − 1).
a. b. c. d.
Slide: 22
24x2 + 24x 2x2 + 2x 13x2 + 9x 13x2 + 35x
Expand and simplify 12x(x + 3) + x(x − 1).
Slide: 23
Evaluate and simplify the following expression: –5a2b3(–2a + ab2 – 4b) + 4a3b3(–4 + 2ab2)
Slide: 24
Basic Arithmetic Operations with Algebraic Expressions 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
Slide: 25
Multiply (2x2 + 3) and (5x + 4).
Multiply (2x2 + 3) and (x2 + 4x + 1). a. b. c. d.
Slide: 26
4x4 + 8x3 + 5x2 + 12x + 3 4x4 + 8x3 + 3x2 + 12x + 5 4x4 + 11x3 + 12x2 + 3x 2x4 + 8x3 + 5x2 + 12x + 3
Multiply (2x2 + 3) and (x2 + 4x + 1).
Slide: 27
Basic Arithmetic Operations with Algebraic Expressions 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
Slide: 28
Divide 12x2y by 9x.
Divide –12a2b2c2 by 24a6b4c2. a.
−𝟏𝒂𝟒 𝒃𝟐 𝟐
b.
−𝟏 𝟐𝒂𝟒 𝒃𝟐
c.
−𝟏𝒂𝟑 𝒃𝟐 𝒄 𝟐
d.
−𝟏 𝟐𝒂𝟑 𝒃𝟐 𝒄
Slide: 29
Divide –12a2b2c2 by 24a6b4c2.
Slide: 30
Basic Arithmetic Operations with Algebraic Expressions 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
Slide: 31
Divide (25x3 + 15x2) by 10x.
Divide (6x3 + 9x4 + 11x) by 9x4.
Slide: 32
Basic Arithmetic Operations with Algebraic Expressions All arithmetic operations can be applied to algebraic expressions.
Division: Dividing Polynomials by Polynomials. When dividing a polynomial by a polynomials, we divide the polynomials using long division, as we do with numbers.
Example
Slide: 33
Divide (8x2 + 2x – 3) by (2x – 1).
Basic Arithmetic Operations with Algebraic Expressions All arithmetic operations can be applied to algebraic expressions.
Division: Dividing Polynomials by Polynomials.
1. 8x2 ÷ 2x = 4x 2. 4x(2x – 1) = 8x2 – 4x 3. (8x2 + 2x – 3) – (8x2 – 4x) = 6x – 3 4. 6x ÷ 2x = 3 5. 3(2x – 1) = 6x – 3 6. (6x – 3) – (6x – 3) = 0
Slide: 34
Evaluate the following expression:
Slide: 35
6x3 – 9x2 + 1 3x2 + 2
Evaluate the following expression:
Slide: 36
6x3 – 9x2 + 1 3x2 + 2
Slide: 1
Section 2.1
Algebraic Expressions
Slide: 2
Section 2.1
Algebraic Expressions 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.
Arithmetic Expressions
13 + 11 56 − 26 12 × 9 25 ÷ 5
13x + 11 56 − 26y 12(9a + 4) 25b ÷ 5
Algebraic Expressions
Letters may be used to represent variables or constants. Slide: 3
Section 2.1
Writing Simple Algebraic Equations An algebraic equation is a mathematical statement that shows the equality between two algebraic expressions or between a number and an algebraic expression.
Eight more than a number, ‘n’, is 19 28 decreased by twice a number is 10
28 – 2x = 10
The product of 7 and the sum of a number and 9 is 77
7(x + 9) = 77
A number divided by 3 is 8
Slide: 4
n + 8 = 19
n =8 3
Variables and Constants 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.
Example
Let’s assume that it is ten degrees hotter in the sun than it is in the shade.
Temperature in the sun.
Temperature in the shade.
y = x + 10 The temperature may vary, so x and y are free to change, but the relationship between them is fixed. Slide: 5
Algebraic Expressions 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. Expression 1st Term
2nd Term
3rd Term
y 2 4x + x − 25 Variable
Slide: 6
Coefficient:
Constant term:
the numerical factor in front of the variable in a term.
a term that has only a number without any variables.
Algebraic Expressions 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. Expression 1st Term
2nd Term
3rd Term
y 2 4x + x − 25 Variable
Slide: 7
Coefficient:
Constant term:
the numerical factor in front of the variable in a term.
a term that has only a number without any variables.
Like Terms and Unlike Terms 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.
Like terms
2x and 43x 21y2, −6y2, and 15y2 6xy and –2yx Slide: 8
Unlike terms
4y and 8y2 x2, x, and 1
Like Terms and Unlike Terms 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.
The distinction is important because like terms can be added or subtracted from each other, while unlike terms cannot.
=
+ x
x
2x
+
=
y
y
+ x Slide: 9
2y
+
= y
x
y
Number of Terms in an Expression 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.
Monomial
Binomial
Trinomial
3
3+x
3 + x + 4y
5x
5x − 2y
5x − 2y + 17xy
9xy
9xy + 21x
9x + 21y − 6z Polynomial
9x + 21y − 6z − 12 Slide: 10
9x + 21y − 6z + 3xyz − 2, and so on…
Evaluating Algebraic Expressions To evaluate algebraic expressions, we replace all the variables with numbers and simplify the expression.
Example
Slide: 11
Evaluate 5x + 2y, where x = 3 and y = 5.
5 x + 2y = 5 (3) + 2 (5) = 15 + 10 = 25
Evaluate the expression and y = 2. a. b. c. d.
Slide: 12
4.33 5.7 4.75 5.2
7xy + 5x 2y + 6
, where x = 3,
Evaluate the expression and y = 2.
Slide: 13
7xy + 5x 2y + 6
, where x = 3,
Basic Arithmetic Operations with Algebraic Expressions 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).
(3x2 + 7x − 1) + (11x2 − 4x + 13) Identify like terms and group them.
= 3x2 + 7x − 1 + 11x2 − 4x + 13 = 3x2 + 11x2 + 7x − 4x + 13 − 1 = 14x2 + 3x + 12
Slide: 14
Subtract (6x2 − 17x + 5) from (14x2 + 3x − 3).
a. b. c. d.
Slide: 15
−8x2 − 20x + 8 20x2 − 14x + 2 8x2 + 20x − 8 8x2 − 14x − 8
Subtract (6x2 − 17x + 5) from (14x2 + 3x − 3).
Slide: 16
Evaluate the following expression: a2 – {–3a + [7 – (5ab + 9a) – 2a2] – 1}
Slide: 17
Basic Arithmetic Operations with Algebraic Expressions 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.
Group coefficients and variables
11x2y × 2xy = (11)(2)(x2)(x)(y)(y) = 22x3y2
Slide: 18
Multiply 9a3, 2ab, and 5b2.
a. b. c. d.
Slide: 19
90a4b3 90a3b2 16a4b3 16a3b2
Multiply 9a3, 2ab, and 5b2.
Slide: 20
Basic Arithmetic Operations with Algebraic Expressions 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
Slide: 21
Multiply 4x2 and (2x2 + 5x + 3).
Expand and simplify 12x(x + 3) + x(x − 1).
a. b. c. d.
Slide: 22
24x2 + 24x 2x2 + 2x 13x2 + 9x 13x2 + 35x
Expand and simplify 12x(x + 3) + x(x − 1).
Slide: 23
Evaluate and simplify the following expression: –5a2b3(–2a + ab2 – 4b) + 4a3b3(–4 + 2ab2)
Slide: 24
Basic Arithmetic Operations with Algebraic Expressions 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
Slide: 25
Multiply (2x2 + 3) and (5x + 4).
Multiply (2x2 + 3) and (x2 + 4x + 1). a. b. c. d.
Slide: 26
4x4 + 8x3 + 5x2 + 12x + 3 4x4 + 8x3 + 3x2 + 12x + 5 4x4 + 11x3 + 12x2 + 3x 2x4 + 8x3 + 5x2 + 12x + 3
Multiply (2x2 + 3) and (x2 + 4x + 1).
Slide: 27
Basic Arithmetic Operations with Algebraic Expressions 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
Slide: 28
Divide 12x2y by 9x.
Divide –12a2b2c2 by 24a6b4c2. a.
−𝟏𝒂𝟒 𝒃𝟐 𝟐
b.
−𝟏 𝟐𝒂𝟒 𝒃𝟐
c.
−𝟏𝒂𝟑 𝒃𝟐 𝒄 𝟐
d.
−𝟏 𝟐𝒂𝟑 𝒃𝟐 𝒄
Slide: 29
Divide –12a2b2c2 by 24a6b4c2.
Slide: 30
Basic Arithmetic Operations with Algebraic Expressions 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
Slide: 31
Divide (25x3 + 15x2) by 10x.
Divide (6x3 + 9x4 + 11x) by 9x4.
Slide: 32
Basic Arithmetic Operations with Algebraic Expressions All arithmetic operations can be applied to algebraic expressions.
Division: Dividing Polynomials by Polynomials. When dividing a polynomial by a polynomials, we divide the polynomials using long division, as we do with numbers.
Example
Slide: 33
Divide (8x2 + 2x – 3) by (2x – 1).
Basic Arithmetic Operations with Algebraic Expressions All arithmetic operations can be applied to algebraic expressions.
Division: Dividing Polynomials by Polynomials.
1. 8x2 ÷ 2x = 4x 2. 4x(2x – 1) = 8x2 – 4x 3. (8x2 + 2x – 3) – (8x2 – 4x) = 6x – 3 4. 6x ÷ 2x = 3 5. 3(2x – 1) = 6x – 3 6. (6x – 3) – (6x – 3) = 0
Slide: 34
Evaluate the following expression:
Slide: 35
6x3 – 9x2 + 1 3x2 + 2
Evaluate the following expression:
Slide: 36
6x3 – 9x2 + 1 3x2 + 2
