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Saturday 25th of February
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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Multiplying by a Monomial

In this document, we revisit the meaning of brackets when used in algebraic expressions.

Even the most complicated applications of brackets can eventually be considered to be an application of the so-called distributive law

a(b + c) = ab + ac

That is, by placing ‘b + c’ inside the brackets and multiplying the bracketed expression by ‘a’, we really mean that every term inside the brackets is to be multiplied by ‘a’. We are distributing the factor ‘a’ to all of the terms inside the brackets.

In this note, we’ll consider situations where ‘a’ is just a single term (a monomial). However, in the next note, we’ll demonstrate how this pattern works when ‘a’ is replaced by a bracketed expression with more than one term itself.

The process of eliminating a pair of brackets as shown in the distributive law above is called expanding the brackets. Later, we’ll look at situations in which it is useful to start with an expression having the pattern of the right-hand side of the distributive law above and rewriting it as a product like the one on the left-hand side. This operation is call factoring.


Example 1:

Expand 3x(5x – 3y + 2) .


Just apply the pattern shown in the distributive law above:

3x(5x – 3y + 2)

= (3x)(5x) + (3x)(-3y) + (3x)(2)

= 15x 2 – 9xy + 6x

Notice we took care to account for the minus sign preceding the second term in the brackets. This is very important.

Often expansion of two or more terms is a useful first step in being able to simplify an algebraic expression.


Example 2:

Simplify: 3x(2x + 5y – 7) – 5y(3x – 4y + 9) + 15x.


So far, we have just one method for simplifying an algebraic expression – namely the collection of like terms. It looks as if there might be like terms in this expression because there’s x’s and y’s all over the place. However, as the expression is written, the three terms

3x(2x + 5y – 7), – 5y(3x – 4y + 9), and 15x

are by no means “like terms.” (Remember, the terms of an expression are always the parts of the expression connected by ‘+’ and ‘-‘ operations.)

However, we can try expanding the terms with brackets to see if any progress results.

3x(2x + 5y – 7) – 5y(3x – 4y + 9) + 15x

= (3x)(2x) + (3x)(5y) + (3x)(-7) + (-5y)(3x) + (-5y)(-4y) + (-5y)(9) + 15x

= 6x 2 + 15xy – 21x – 15xy + 20y 2 – 45y + 15x

= 6x 2 + (15 – 15)xy + 20y 2 + (-21 + 15)x – 45y

= 6x 2 + 20y 2 – 6x – 45y

as the final simplified answer.

So, when the bracketed terms were expanded, like terms were found in the result. Notice the care we took above to account for subtraction and minus signs.


Example 3:

Expand: -5y 2 (3xy – 5y – 9x 2 y) .


The real issue here is to keep careful track of all of the minus signs. In detail, we can do the work as follows:

-5y 2 (3xy – 5y – 9x 2 y)

= (-5y 2 )(3xy) + (-5y 2 )(-5y) + (-5y 2 )(-9x 2 y)

= -15xy 3 + 25y 3 + 45x 2 y 3

To cope with the minus signs here, we regarded them as attached to their terms – in effect converting subtractions to additions of negative terms:

3xy – 5y – 9x 2 y = 3xy + (-5y) + (-9x 2 y)

This is often a useful strategy when an expression to be simplified or manipulated has many negated or subtracted terms.

Remember that a minus sign outside of a set of brackets can be considered equivalent to multiplication by -1.

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