Will my answer always be in parenthesis?

Question

Like factoring a difference between two squares

Answer 1

Hi Rayven,

 

In the case of factoring the difference between two squares, the answer is: yes, the answer will always have two sets of expressions in parentheses. This is because in factoring you are breaking down a more complicated expression into simpler parts that, when multiplied together give you your original more complicated expression. The parentheses contain each of the simpler parts, and when we write a number or other more complicated expression next to another expression in parentheses that means multiplication.

 

So the parentheses serve two functions. One, containing the simpler parts; two, showing that the two parts are meant to be multiplied.

 

So, in the case of factoring two squares, the formula is:

 

x² - y² = (x + y)(x - y)

 

If you multiply out the part to the left of the equals sign, you can see how it becomes the part on the right:

 

(x + y)(x - y) = x(x - y) + y(x - y)      [distribute]

x(x - y) + y(x - y) = xx - xy + xy - yy [distribute again]

xx - xy + xy - yy = x² - xy + xy - y²   [just writing xx as x² and yy as y²]

x² - xy + xy - y² = x² - y²                 [ cancel: -xy + xy = 0]

 

So, that shows that (x + y)(x - y) = x² - y² which is what we wanted.

 

The larger point is that the parentheses contain the factors, and putting the factors in parentheses right next to each other means to multiply them.

 

Not every answer to questions involving things besides the difference of two squares will contain parentheses, but many will. In general you use parentheses to group together a set of terms which will then be multiplied by another factor, or to group together terms for other reasons.

 

Answer 2

Parentheses are used because they are the highest order of mathematical operations. They are only necessary to override when a lesser order of operation needs to be performed prior to what the normal order of operations would dictate being performed. For example:

X + Y² is different than (X + Y)²

The first one says to square Y and add it to X.

The second one says to sum X and Y and then square the sum.

Answer 3

Are you meaning like if you have:

 

(x² - 4)

 

And your answer is:

 

(x - 2)(x + 2)

 

Then yes.

 

This means the result of (x - 2) times the result of (x + 2).  Since we don't know what the result of either of those are (since we don't know what x is), they have to stay in parenthesis.  This is showing we're doing the subtraction inside the one, and the addition inside the other, before multiplying those answers.

 

So the parenthesis are necessary for proper order.

 

(Of course, that only addresses your one example.  There are times when parenthesis aren't going to be necessary, but they are in this case.)