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# Simplifying rational expressions: common binomial factors

Given a rectangle with length a²+6a+27 and width a²-9, Sal writes the ratio of the width to the length as a simplified rational expression. Created by Sal Khan.

## Want to join the conversation?

• Is (x+3)/(x+9) our final answer? • for the final answer at the end of the video a+3/a+9 couldn't you divide 3 and 9 so the final answer is a+1/a+3? • No, it's a bit more complicated. Think about it this way:

You want to simplify the fraction. One way to do this would be dividing by 3. So you have:

( ( a + 3 ) / ( a + 9 ) ) / 3

You could then distribute the 3 over the numerator ( a + 3 ) and denominator ( a + 9 ):

( ( a + 3 ) / 3 ) / ( ( a + 9 ) / 3 )

Now you have to distribute the 3 across each of the terms ( a + 3 ) and (a + 9 ):

( ( a / 3 ) + ( 3 / 3 ) ) / ( ( a / 3 ) + ( 9 / 3 ) )

Simplifying, you get:

( ( a / 3 ) + 1 ) / ( ( a / 3 ) + 3 )

Of course, this isn't a very simple way to express ( a + 3 ) / ( a + 9 ).
• I'm still kind of confused on the values that the variable cannot be equal to. For example a can't be 3 or -9. Could anybody answer that area for me? Thanks a lot! • Derek,
If x=-9 then the denominator would be 0
and if x=3 then the denominator would also be 0
And if the denominator is 0 the answer is undefined.
So the domain cannot include -9 or 3.

After factoring and reducing the expression by making the (a-3)/(a-3) = 1
this left only (a+9) in the denominator.
But because the original equation did not allow a to be -9 or 3, the factored and reduced equation also cannot be equal to -9 or 3.

I hope that helps.
• Near the end It first was a not equal to -9 or 3. Then when only (a+3) / (a+9) was left, he wrote: a is not equal to -9 or 3. But if you fill in a=3. That would make a (3+3= )6 and not 0. Or am I missing something? I thought the answer at the final should be a -3 or -9. • In the final expression a+3/a+9, i see that if a= 3 then that would result in a defined answer which is 6/12. That can be simplified to 3/4, too. So why "a" cannot be equal to 3 at the end of the video? • what if after i factor everything out, but nothing is able to be cancled out, would that mean that it is already in simplest form? • is there a difference between rationalising something and simplifying a rational expression? • If the width is over the length, does that always have to be the case? Can't you flip them, or is that illegal math? • Noah,
A ratio of two things can be writtern either way. It is perfectly acceptable (and often necessary to flip them. The only reason it would be restricted is because the problem required an answer in a specific manner.

For instance, if you have a ratio of 12 miles to 1 hour, It could be written as
12miles/1 hour or
1 hour/12 miles.

And depending on the problem you would need to figure out which way to use the ratio.

For instance if you had a problem where you were give that a person traveled by bike at a ratio of 12 miles to 1 hour and he had traveled for 2 hours. You are asked how far he had traveled. You would use the equation.
2 hour * 12 miles/1hour = 24 miles
Notice the words (hours* miles/hours) results in hours in both the numerator and denominator whcih canel out giving you an answer in miles.

But is you were given a problem that the person needed to travel 36 miles and you were asked how long it would take, you would use the equation
36 miles * 1 hour/12 miles = 3 hours
And notice this time the words (miles*hour/miles) results in miles in both the numerator and denominator which cancel out giving you an answer in hours.

In one equation we used the ratio one way and in the other we flipped it over.

The trick to using ratios is to realize they work either way and you need to construct an equation so that the words cancel out the correct way to give you the correct words in your answer.  