Explanation:
we know, the area of a rectangle is
length × width
so, we have here
length × (x - 5) = x² - 7x + 10
therefore,
length = (x² - 7x + 10) / (x - 5)
so, just by looking at it (with experience) I see that length is (x - 2).
but let's do the actual polynomial division, which seems to be the desired exercise here.
x² - 7x + 10 ÷ x - 5 =
it is very similar to actual number divisions.
we go from left to right.
we start with the first term (x²) and see how many times (x - 5) would fit into it. for that we look only at the first term in x - 5 (that is x).
x² / x = x
that is already our first term of the result (quotient) :
x² - 7x + 10 ÷ x - 5 = x
now, we multiply that result with the divisor (x - 5) and subtract that from the dividend.
x² - 7x + 10 ÷ x - 5 = x
- x² - 5x
--------------
0 - 2x
and now we pull down the next term from the dividend :
x² - 7x + 10 ÷ x - 5 = x
- x² - 5x
--------------
0 - 2x + 10
and we repeat the process with that new dividend :
-2x / x = -2
that is our next term of the quotient.
x² - 7x + 10 ÷ x - 5 = x - 2
- x² - 5x
--------------
0 - 2x + 10
we repeat the process of multiplying that term (-2) with the divisor (x - 5) and subtracting that result from the dividend :
x² - 7x + 10 ÷ x - 5 = x - 2
- x² - 5x
--------------
0 - 2x + 10
- -2x + 10
-------------
0 0
there is no remainder, and there is no more term to pull down. we are finished.
the result of the division is (as expected) : x - 2
the expression to describe the length is therefore
x - 2