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Ashley- what are you supposed to do with this equation? i.e. does it ask you to solve for f, factor, graph, etc. ?

I don't know where to start with this either. What was the prompt for all of the problems in the section this was under?

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Knowing that I don't see an equal sign on this polynomial, it is meant that it must be treated as an expression, so I assume the objective with this expression is to factor it.

If factoring is the objective in this exercise, consider there are two areas of interest in this expression:

=> First, c2 - 2c + 1 is a perfect square trinomial, so it is factored this way:

      c2 - 2c + 1 = c2 - c  - c + 1              

                      = c(c - 1) - 1(c - 1)

                      = (c - 1)(c - 1)

                      = (c - 1)2

Note: The above trinomial was factored by grouping.

Second, the term 9f 2 can be written as (3f)2 knowing that both 9 and f 2 are squares.

With both areas of interest being covered, the original expression can be rewritten as (c - 1)2 - (3f )2.

Notice that the rewritten expression has the following notation: x2 - y2, which is a difference of squares. In this case, x = (c - 1)2 and y = (3f)2.

Since, for any difference of squares, x2 - y2 = (x - y)(x + y), the expression

(c - 1)2 - (3f)2 = [(c - 1) - 3f] [(c - 1) + 3f].

The use of brackets was needed to distinguish one factor from the other.