CONFIDENTIAL 1 Grade 8 Algebra I Introduction to Algebra.

CONFIDENTIAL 1

Grade 8 Algebra IGrade 8 Algebra I

Introduction to Introduction to AlgebraAlgebra

CONFIDENTIAL 2

1) 2 = 7 - x

2) - 3b = 21

3) 2x + 6 = 14

4) 4(9 - x) = 16

5) 4 - 3x = -8

Warm UpWarm Up

Solve the equation:

CONFIDENTIAL 3

Introduction

Constants and variable: A symbol in algebra having a fixed value is called a constant, whereas a

symbol which can be assigned different values is called a variable.

Example: 1/3, -8, Π,√2 are all constants and x, y, z are all variables.

Algebraic expression: A combination of constants and variables connected by signs +, -, × and ÷ is

called an algebraic expression. The several parts of an expression separated by + or - sign are called the

terms of an expression.

Here is a typical polynomial:

9x2 + 2x - 5

Terms

CONFIDENTIAL 4

Monomial: is a variable that is formed with a number and a letter variable to its powers.

Example: 3x3 is a monomial.

You can’t add or subtract monomials if they have different exponents such as 3x3 and 4x4.

But you can multiply or divide them. To multiply monomials, just add the exponents of the variables and multiply the coefficients.

3x3 x 4x4 = 12x7.

Here are some additional ways to manipulate the monomials:

(am)n = amn

(ab)m= ambm

CONFIDENTIAL 5

Polynomial: is a mathematical expression involving a sum of powers in one or more

variables multiplied by coefficients.Example: 7x4 + 6x2 + x is a polynomial .

Each piece of the 7x4 + 6x2 + x , each part that is being added, is called a "term".

When a term contains both a number and a variable part, the number part is called the "coefficient". The coefficient

on the leading term is called the leading coefficient.

9x2 + 2x - 5

Leading coefficient is 9.

coefficient

CONFIDENTIAL 6

6x –2 NOT a polynomial term.

This has a negative exponent.

1

NOT a polynomial term.

This has the variable in the denominator.

sqrt(x) NOT a polynomial term.

This has the variable inside a radical.

4x2 A polynomial term .

Here are some examples:

x2

Polynomial terms have variables to whole-number exponents; there are no square roots of exponents, no fractional powers, and no variables in the denominator.

CONFIDENTIAL 7

Degree: If the polynomial is of one variable, then the highest power of the variable is called

the degree if the polynomial.

7x4 + 6x2 + x is a polynomial in x of degree 4.

Linear polynomial: is a polynomial of degree 1.Example: (4x + 2), (3/5 + 7x) is a linear polynomial .

An equation says that two things are equal. It will have an equals sign "=" like this: Example: 4a + 5 =13 is an equation.

CONFIDENTIAL 8

ADDITION AND SUBTRACTION OF POLYNOMIALS:

1) Collect the like terms together.

2) Find the sum or difference of the numerical coefficients of these terms.

3) The resulting expression should be in the simplest form and can be written according or descending order of terms.

CONFIDENTIAL 9

Add: (4y2 + y - 6) + (6y2 - 4y + 9)

= (4y2 + 6y2) + (y - 4y) - 6 + 9= 10y2 - 3y + 3

Subtract: (9y2 + 3) - (5y2 + 6)

= (9y2 - 5y2) + 3 - 6= 4y2 - 3

CONFIDENTIAL 10

Add:

1) 6y + 9x and 4a - 7x

2) 4y2 + 5y and 4y2 - 3y

Subtract:

4) -7x2y from -9x2y

3) -5x2y2 + x2y2 , 7x2y2 , x2y2-11 5

23

5) 9y2 + 3 from 5y2 + 6

6)  The degree of the polynomial 7y4 - 10y is _______.

Now you try!

CONFIDENTIAL 11

7) Find the width of the rectangular field, if the perimeter is 16y + 8 and the length is y + 4.

8) The two sides of a triangle are 2y and 4y. Perimeter of the triangle is (8y + 1). Find the third side.

Solve:

Now you try!

9)Which of the following expressions is a monomial?• 7y• 4 - y• y2 + 4y - 7

CONFIDENTIAL 12

MULTIPLICATION OF POLYNOMIALS:

Case 1: Multiplication of monomials

Product of monomials = Product of numerical coefficient x Product of literal factors

Case 2: Multiplication of polynomials

Multiply each term of one polynomial with each term of the other polynomial and simplify by taking the like terms together.

CONFIDENTIAL 13

Multiply: 16x2y3 by 4x4y3z

= (16 x 4) ×(x2y3 × x2y3z) = 64x6y6z

Multiply: 4x2 - 6x + 5 by 3x + 2

= 3x(4x2 - 6x + 5) + 2(4x2 - 6x + 5)= (12x3 - 18x2 + 15x) + (8x2 - 12x + 10)= (12x3 - 10x2 + 3x + 10)

CONFIDENTIAL 14

Multiply:

1) (9y2 + 3) and (5y2 + 6)

Now you try!

2) 4y2 + 5y and 4y2 - 3y

3) 6y + 9x and 4a - 7x

CONFIDENTIAL 15

DIVISION OF POLYNOMIALS:

Case 1: Division of monomial by a monomial

Quotient of two monomials = Quotient of numerical coefficient x Quotient of literal factors

Case 2: Multiplication of polynomial by a monomial

Divide each term of the polynomial by the monomial.

CONFIDENTIAL 16

32x3y3 by -8xy

= 32x3y3 = 32 × (x3y3) -8xy -8 ( xy)

= -4x2y2

18x4y2 + 15x2y2 - 27x2y by -3xy

= 18x4y2 + 15x2y2 - 27x2y -3xy

= 18x4y2 + 15x2y2 - 27x2y -3xy -3xy -3xy

= -6x3y - 5xy + 9x

Divide:

CONFIDENTIAL 17

Divide: (14x2 -53x + 45) by (7x - 9)

7x - 9 14x2 -53x + 45

2x - 5

-(14x2 -18x)

-35x + 45

-(-35x + 45)0

CONFIDENTIAL 18

Divide:

Now you try!

1) -81a5b4c3 by -9a2b2c

2) (3ab2c3 - 2a2b2c2 + ab2c) by (abc) 4 5 3 2

3) (x3 -4x2 + 7x - 2) by (x - 2)

CONFIDENTIAL 19

Algebraic Identities

Consider the statement (x + 3)2 = x2 + 6x + 9.

x LHS RHS

1 16 16

2 25 25

3 36 36

4 49 49

On putting various values of x, you find that L.H.S = R.H.S. for all values of x. Such a mathematical sentence containing an unknown variable x which is satisfied for all values of x is

called an identity.

CONFIDENTIAL 20

IDENTITY 1: (x + a) (x + b) = x2 + (a + b)x + ab

IDENTITY 2: (x + a)2 = x2 + 2ab + b2

IDENTITY 3: (x - a)2 = x2 - 2ab + b2

IDENTITY 4: (a + b) (a - b) = a2 - b2

Other formulas:

1) a2 + b2 = (a + b)2 - 2ab

2) a2 + b2 = (a - b)2 + 2ab

3) (a - b)2 = (a + b)2 - 4ab

4) (a + b)2 = (a - b)2 + 4ab

5) (a + b)2 + (a - b)2 = 2(a2 + b2 )

Some identities

CONFIDENTIAL 21

Here are some examples:

1) (a + 3)2 = a2 + 2×a×3 + 32 = a2 + 6a + 9

2) (b - 5)2 = b2 - 2×b×5 + 52 = b2 - 10b + 25

3) (2x + 3)(2x - 3) = (2x)2 - 32 = 4x2 - 9x

4) Find a2 + b2 when (a + b) = 7 and ab = 12

a2 + b2 = (a + b)2 - 2ab

= (7)2 - 2×12

= 49 - 24 = 25

CONFIDENTIAL 22

Now you try!

1) (3a + 7)2

2) (2y + 5)(2y - 5)

3) (2m - 3n)2

4) 3a + b 2

b 2a

Solve:

5) Find a2 + b2 when (a - b) = 5 and ab = 14

CONFIDENTIAL 23

Factorize:

1) a2 + 4a + 4 = (a)2 + 2.2.a + (2)2 = (a + 2)2

2) 9a2 - 4b2

= (3a)2 - (2b)2 = (3a + 2b) (3a - 2b)

3) a2 - b2 = a 2 - b 2 = a + b a - b 25 36 5 6 5 6 5 6

CONFIDENTIAL 24

Factorize:

Now you try!

1) 64a2 + 80ab + 25b2

2) a2 - 2 + 1 a2

3) 16a2 - 4b2 25b2 100a2

CONFIDENTIAL 25

1) The volume of a cube is (5x3 + 8) m3. A smaller cube with volume (x3 + 5) m3 is cut out of the cube.

Choose a polynomial for the remaining volume. 

Now you try!

2) Two square playgrounds have areas 4x2 and 9y2. Express the difference in area in a factored form.

3) Which of the numbers out of 3 and 4 is a solution of the equation, 3x + 2 = 14?

CONFIDENTIAL 26

BREAK

CONFIDENTIAL 27Double-click to edit

CONFIDENTIAL 28

Assignments

1) Add : 8y3 + 4y2 - 8y + 5 and - 8 - 4y3 + 5y2 + 7y.

2) Subtract: 7y - 4y2 - 6 from 2y + 3 - y2.

3) Simplify: (2y + 5)(2y - 5)

4)Which of the following is an equation?I. 2x - 3II. 2y + 2 < 4III. x - 3 = 0

CONFIDENTIAL 29

5) The degree of the polynomial 9y8 + 1 is_____.

6) Which of the numbers out of 2 and 3 is a solution of the equation, 6x + 5 = 23?

7) Find 3y + 4z, for y = 6 and z = 3.

8) Robert and his friends went to a magic show with their families. Cost of ticket for each adult was $25 and

for each child was $12. The group had 14 adults and the total cost for the tickets was $518. How many

children were there in the group?

CONFIDENTIAL 30

 9) Ashley's salary 3 years ago was $y. Now, she gets 4 times the salary and spends $2,598.

Calculate her savings.

10) Factorize: 15x4 - 3x2 - 18

11) Factorize: 1 - 121 a2

12) Factorize: 2x2 + 7xy - 15y2

CONFIDENTIAL 31

You did a great job You did a great job today!today!