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Solving Equations I-Expressions Involving Power Functions

Toolbox—Combining Like Terms

Example 2.1 Simplify the expression (x +1)y + 4xy by combining like terms.

We can deconstruct the left part of the sum into the product of x +1 and y. One way to
deconstruct the right part of the sum is to split it into the product of 4x and y. Using this
deconstruction, we see that y is a common term to each product, so (x +1)y and 4xy
are like terms. To combine them, we factor out the common term and add or subtract
whatever is left over. Here the leftover pieces of the like terms are expressions involving
variables themselves—not just numbers. So what!

identify the common term,
factor out the common term,
add the leftovers.

Example 2.2 Simplify the expression by combining like terms.

This expression may not, at first glance, seem to have any like terms. But if you look
back at Solution 2 of Example 1.3, both the left and right pieces of the sum have a
common factor of . We use the same method as in the first two examples
above:

 
identify the common term,
factor out the common term,
add the leftovers.

Example 2.3 Simplify the expression by combining like
terms.

We have to play with this expression a little bit to find out which deconstruction will be
most useful to us. Right now you may be having trouble knowing the difference between
what is useful and what is practically useless. What we optimally want is to identify all
the common terms.
A useless way: Deconstruct the left-hand piece of the sum into the product of and
2x. Split the right hand piece of the sum into the product of and This
deconstruction, though a perfectly legal (mathematically anyway) way to see the
expression, will not help us at all since we have not identified any common terms in the
two products. Don’t give up—try again!
A useful way: Look at the left-hand piece of the sum as the product of x and
Turn the right hand part of the sum into the product of x and We have
now found one common term—the x—so this is a good start.

An even more useful way: Notice that the leftover terms in the previous way each contain
the component x +1, but raised to different powers. Use this! Since
, we can turn the left-hand part of the sum into the product of
and 2(x +1). The right hand part of the sum can be written as the product of
and . Thus we still see the common term of x, but also the common term
of .

We conclude the example by using this last deconstruction.

 
identify the common term,
factor out the common term,
add the leftovers.
 

Note. In the preceding example, we stopped identifying common terms after we found
the common term . Could we have found any other common terms? How can
you tell when you are done?

You can tell by looking at the leftover terms. In the example, 2(x +1) is leftover from
the left-hand part of the sum, and is leftover from the right. Convince yourself that
these leftover terms cannot be split into products that have a common term.

Common Mistake. A common error here is to think that x is a common term to each, and
that we could factor out the x from both 2(x +1) and . But we cannot write
2(x +1) as x ยท something. The ‘+1’ traps the x.

Toolbox—Zero Law

So what is the difference between an expression and an equation? The simple answer is
that an equation has an ‘=’ symbol. This symbol sets up a relationship between two
expressions—our goal is to find out where this relationship is true. What this means is
that our job is to find out for which values of the variable is the left-hand expression
equal to the right-hand expression.

The Zero Law is a way to turn more complex equations into two or more simpler ones.
Let ex1 and ex2 represent expressions (like x +1 or x2 - 7x + 6). If , for a
particular value of x, then either that value of x makes or it makes (or
possibly both). This is very helpful to us because looking for the solutions of
is the same as looking for the solutions of and separately. Of
course we can do this with any number of expressions which are multiplied together.

Note that the Zero Law only works if one side of the equation is already zero! (Thus the
name…)

Example 2.4 Use the Zero Law to find the solutions of (x +1)(x - 2)(x -1) = 0.

The expression on the left side is the product of three expressions. The Zero Law tells us
that finding which x make the left-hand side equal to zero is the same as finding which x
make at least one of the three pieces on the left equal to zero. We have:

The Zero Law tells us we can solve instead the three equations