factor 1-r^2??

Question

This problem is confusing me

Answer 1

The expression:

     1-r^2

is the difference of two squares:  1^2 and r^2.

 

While it is true that (1+r)(1-r) produces (1-r^2) 

  it is also true that (-1-r)(r-1) produces (1-r^2)

 

Now, are those factors really the same or are they unique?  We might have to test a value of r to assure ourselves.  Let's try:

    r = 1/4

 

  (1-r^2) = (1+r)(1-r)

  (1-(1/4)^2) = (1+(1/4))(1-(1/4))

  (1- (1/16)) = (5/4)(3/4)

       15/16 = 15/16

 

  (1-r^2) = (-1-r)(r-1)

  (1-(1/4)^2) = (-1-(1/4))((1/4)-1)

  (1-(1/16)) = (-5/4)(-3/4)

       15/16 = 15/16

 

So, the value (15/16) has two sets of factors:

        (5/4) and (3/4)

    or

        (-5/4) and (-3/4)

 

There is a HUGE difference when the problem asks you to factor:

   1-r^2 = 0

because 0 is neither positive nor negative and we must set either:

     1-r = 0

  or

    1+r = 0

  because this problem states that the product is 0 and because the Multiplicative Property of Zero requires that one of the factors equals 0.  Thus either r=1 or r=-1.  [note: other values of a and b are not important if SQRT(a)=±SQRT(b)].

 

So, if the problem is an equation (set equal to 0), factor it the usual way; if the problem is an expression, factor it differently.

 

Let's check this by doing the example problem again:

   Solve:    1 - r^2 = 15/16

                 -r^2 = 15/16 - 1

                 -r^2 = -1/16

                  r^2 = 1/26

                  r = ±1/4,    then answer, what is (1-r) and (1+r)?  Are there two sets of answers?

 

                1 - r^2 = 0

                  -r^2 = -1

                   r^2 = 1

                   r = ±1     then answer, what is (1-r) and (1+r)?  Are there two sets of answers?

Answer 2

Rowan, this is the difference of two squares!

 

a² - b² = (a - b)(a + b)

 

1 - r² = (1 -r)(1 + r)

 

Hope this helps!

Answer 3

David is correct :) Since 1 has a whole number square root (1) and r² has a whole number square root (r) and they're being subtracted, it is called a difference of squares. I'm not 100% sure that r is a whole number but r is the square root of r² so that's all we need to know. :)