So... I know the answer to the question because it's something we worked on in class... but there's one part I don't get.
Factoring
m - 2n + m^2 - 4n^2
m - 2n + (m + 2n) (m - 2n)
Then there's a 1...
1(m - 2n) (m + 2n) (m - 2n)
AND my question is...
How and why does this 1 go over here and do this from the step above?? ...
(m - 2n) (1 + m + 2n)
And the answer somehow is
(m - 2n) (1 + m + 2n)
m - 2n + (m + 2n) (m - 2n)
(m - 2n) + (m + 2n) (m - 2n)
This is like
a + b a
a(1+b) where a is m-2n and b is m+2n
good
To understand how the 1 appears in the factored expression, let's break down the steps of factoring:
Step 1:
Given the expression m - 2n + m^2 - 4n^2, we want to factor it. We observe that the terms m - 2n and m^2 - 4n^2 can be factored separately.
Step 2:
We start by factoring m - 2n. This expression does not have any common factors, so it remains the same.
Step 3:
Next, we factor m^2 - 4n^2. This is a difference of squares, so we can use the formula a^2 - b^2 = (a + b)(a - b). In this case, a = m and b = 2n. Applying the formula, we get (m + 2n)(m - 2n).
Step 4:
Now, we combine the factored expressions from step 2 and step 3. We have m - 2n + (m + 2n)(m - 2n).
Step 5:
To simplify further, we can distribute the addition to both terms in the expression (m + 2n)(m - 2n). This yields m - 2n + m(m - 2n) + 2n(m - 2n).
Step 6:
On simplifying, we get m - 2n + m^2 - 2mn + 2mn - 4n^2.
Step 7:
Now, notice that the terms -2mn and +2mn cancel each other out, leaving us with m - 2n + m^2 - 4n^2.
Step 8:
At this point, we introduce a 1 as a common factor in the expression m - 2n. This is just multiplying by 1, so the expression remains the same. We get 1(m - 2n) + m^2 - 4n^2.
Step 9:
Finally, we group the terms within parentheses and obtain (m - 2n)(1 + m^2 - 4n^2).
Step 10:
Now, we notice that m^2 - 4n^2 is a difference of squares, once again. It can be factored as (m + 2n)(m - 2n). Substituting this in, we get (m - 2n)(1 + m + 2n).
Therefore, the factored expression is (m - 2n)(1 + m + 2n). This is equivalent to (m - 2n) (1 + m + 2n), where the 1 comes from factoring out a common factor.