algebra

The Binary Operations

Describe the binary operations

When two numbers are combined according to the instructions given and produce one number we say that they are binary operations. For example, when we add 4 to 6 we get 10, or when we multiply 4 by 3 we get 12. We see that addition of two numbers lead to one number and multiplication of two numbers produce one number. This is binary operation. The instructions can be given either by symbols like X,*,∇and so on or by words.

Performing Binary Operations

Perform binary operations

Example

Perform binary operations

solution

Example

Perform binary operations

solution

Example

Perform binary operations

Example

Perform binary operations

Example

Perform binary operations

Brackets in Computation

Brackets are used to group items into brackets and these items inside the brackets are considered as whole. For example,15 ÷(X + 2) ,means that x and 2 are added first and their sum should divide 15. If we are given expression with mixed operations, the following order is used to perform the operations: Brackets (B) are opened (O) first followed by Division (D) then Multiplication (M), Addition (A) and lastly Subtraction (S). Shortly is written as BODMAS.

Basic Operations Involving Brackets

Perform basic operations involving brackets

Example

Perform basic operations involving brackets

4 + 2b – (9b ÷3b)
4z – (2x + z)

solution

Algebraic Expressions Involving the Basic Operations and Brackets

Simplify algebraic expressions involving the basic operations and brackets

Example

Simplify algebraic expressions involving the basic operations and brackets

solution

Identities

For example, 3(2y + 3) = 6y + 9, when y = 1, the right hand side (RHS) and the left hand side (LHS) are both equals to 15. If we substitute any other, we obtain the same value on both sides. Therefore the equations which are true for all values of the variables on both sides are called Identities. We can determine whether an equation is an identity or not by showing that an expression on one side is identical to the other expression on the other side.

Example

Simplify algebraic expressions involving the basic operations and brackets

solution

Quadratic Expressions

A Quadratic Expression from Two Linear Factors

Form a quadratic expression from two linear factors

A quadratic expression is an expression where the highest exponent of the variable (usually x) is a square (x2). It is usually written as ax2+bx+c.

Activity

Quadratic expression from two linear factors

The General Form of Quadratic Expression

Write the general form of quadratic expression

Quadratic expression has the general form of ax2 + bx + c where a ≠ 0 and a is a coefficient of x2 , b is a coefficient of x and c is a constant. its highest power of variable is 2. Examples of quadratic expressions are 2x2 + x + 1, 4y2 + 3, 3z2 – 4z + 1 and so on. In a quadratic expression 3z2 - 4z + 1, a = 3, b = -4 and c = 1. Also in quadratic expression 4y2 + 3, a = 4, b = 0 and c = 3

Example

Write the general form of quadratic expression

Solution

Example

Write the general form of quadratic expression

Factorization

Linear Expressions

Factorize linear expressions

The operation of resolving a quantity into factors, when we expand expressions, is done by removing the brackets. The reverse operation is Factorizing and it is done by adding brackets.

Example

Factorize linear expressions

Solution

In factorization of 5a+5b, we have to find out a common thing in both terms. We can see that the expression 5a+5b, have got common coefficient in both terms, that is 5. So factoring it out we get 5(a+b).

Example

Factorize linear expressions

Solution

Factorizing 18xyz-24xwz, we have to find out highest common factor of both terms. Then factor it out, the answer will be 9xz(2y-3w).

Quadratic Expressions