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 socalled 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
righthand side of the distributive law above and rewriting it as
a product like the one on the lefthand 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.
