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Saturday 15th of June
Rotating a Parabola
Multiplying Fractions
Finding Factors
Miscellaneous Equations
Mixed Numbers and Improper Fractions
Systems of Equations in Two Variables
Literal Numbers
Adding and Subtracting Polynomials
Subtracting Integers
Simplifying Complex Fractions
Decimals and Fractions
Multiplying Integers
Logarithmic Functions
Multiplying Monomials
The Square of a Binomial
Factoring Trinomials
The Pythagorean Theorem
Solving Radical Equations in One Variable
Multiplying Binomials Using the FOIL Method
Imaginary Numbers
Solving Quadratic Equations Using the Quadratic Formula
Solving Quadratic Equations
Order of Operations
Dividing Complex Numbers
The Appearance of a Polynomial Equation
Standard Form of a Line
Positive Integral Divisors
Dividing Fractions
Solving Linear Systems of Equations by Elimination
Multiplying and Dividing Square Roots
Functions and Graphs
Dividing Polynomials
Solving Rational Equations
Use of Parentheses or Brackets (The Distributive Law)
Multiplying and Dividing by Monomials
Solving Quadratic Equations by Graphing
Multiplying Decimals
Use of Parentheses or Brackets (The Distributive Law)
Simplifying Complex Fractions 1
Adding Fractions
Simplifying Complex Fractions
Solutions to Linear Equations in Two Variables
Quadratic Expressions Completing Squares
Dividing Radical Expressions
Rise and Run
Graphing Exponential Functions
Multiplying by a Monomial
The Cartesian Coordinate System
Writing the Terms of a Polynomial in Descending Order
Quadratic Expressions
Solving Inequalities
Solving Rational Inequalities with a Sign Graph
Solving Linear Equations
Solving an Equation with Two Radical Terms
Simplifying Rational Expressions
Intercepts of a Line
Completing the Square
Order of Operations
Factoring Trinomials
Solving Linear Equations
Solving Multi-Step Inequalities
Solving Quadratic Equations Graphically and Algebraically
Collecting Like Terms
Solving Equations with Radicals and Exponents
Percent of Change
Powers of ten (Scientific Notation)
Comparing Integers on a Number Line
Solving Systems of Equations Using Substitution
Factoring Out the Greatest Common Factor
Families of Functions
Monomial Factors
Multiplying and Dividing Complex Numbers
Properties of Exponents
Multiplying Square Roots
Adding or Subtracting Rational Expressions with Different Denominators
Expressions with Variables as Exponents
The Quadratic Formula
Writing a Quadratic with Given Solutions
Simplifying Square Roots
Adding and Subtracting Square Roots
Adding and Subtracting Rational Expressions
Combining Like Radical Terms
Solving Systems of Equations Using Substitution
Dividing Polynomials
Graphing Functions
Product of a Sum and a Difference
Solving First Degree Inequalities
Solving Equations with Radicals and Exponents
Roots and Powers
Multiplying Numbers
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Use of Parentheses or Brackets (The Distributive Law)

The Basic Idea

Pairs of parentheses or brackets (such as ( ), [ ], { }, etc) are one way that is used to group parts of an expression together to show exactly in what order the arithmetic is to be done. Thus, for example, when we write

2(3 + 5)

we mean “add 3 to 5 first, then multiply the result by 2.” Thus

2(3 + 5) = 2 x 8 = 16 (*)

This expression can also be interpreted to mean that the multiplication by 2 is to be done to every term inside the brackets. Thus

2(3 + 5) = 2 x 3 + 2 x 5 = 6 + 10 = 16 (**)

which is the same final result as before. In the sequence of steps in (**), we say that we are expanding the brackets. If we use the symbols a, b, and c to represent any three numbers, then the overall process in (**) can be symbolized as

a(b + c) = a × b + a ×c

The rule in the box above is called the distributive law for multiplication. It shows how multiplication of the bracketed expression by ‘a’ is “distributed” to all of the terms in the brackets.

Be careful if there are minus signs inside the brackets. For example

2(3 – 5) = 2(-2) = -4


2(3 – 5) = 2 x 3 + 2 x (-5) = 6 + (-10) = 6 – 10 = -4

Thus, the minus sign must be retained when the brackets are removed. In general

a(b - c) = a × b - a × c


An Illustration

You can visualize the equivalence between

2(3 + 5) and 2 × 3 + 2 × 5

by considering the following diagrams. We draw 16 dots arranged in two rows as follows:

Each row has a group of three dots and a group of 5 dots, so the number of dots in each of the two rows is 3 + 5 = 8 dots. Now we can count the total number of dots in the diagram using the following scheme:

That is, to get the total of 16 dots in the diagram, we count 8 dots in each of two rows, for a total of 16 dots:

2(3 + 5) = 2 × 8 = 16.

Alternatively, we could imagine grouping the dots in a different way for counting:

This last diagram counts the dots according the process

2 × 3 + 2 × 5 = 6 + 10 = 16.

By comparing the last two diagrams, you can see how the equivalence between the two expressions

2(3 + 5) and 2 × 3 + 2 × 5

comes about. In this illustration, the two expressions just amount to counting the same array of 16 dots, but tallying the dots in different orders.


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