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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) .

solution:

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.

solution:

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) .

solution:

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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