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# Simplifying rational expressions: opposite common binomial factors

Sal simplifies and states the domain of (x^2-36)/(6-x). Created by Sal Khan.

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• why did the negative one only go to a part of the numerator instead of being distributed onto the entire one? • Do you mean that in (x+6)(x-6)(-1)(-1), why should -1 only multiply to (x-6) but not (x+6)(x-6)?

If that's what you mean, maybe I know the answer of this question.

Do you know the equation "a×b×c×d = a×(b×c)×d"? That's the equation that uses here. Imaging that (x+6) is "a", (x-6) is "b", the first -1 is "c" and the second -1 is "d".

In multiplication, the numbers could change order and wouldn't change the result. Like "1×2×3×4 = 2×3×1×4".And that's why the -1 only multiply to (x-6), but not the (x+6)(x-6).

But if it's in addition, let's say the if it's (a+b)×c×d, you could say that it's equal to (a×c+b×c)×d. The reason why that's fine to change it like this is because (a+b)×c×d = (a+b)×(c×d) = a×(c×d)+b×(c×d) AND (a×c+b×c)×d = a×c×d+b×c×d.

Hope that helps!
• I was working on the problem set for this, and found that even if the denominator turned out to be something like (y + 5)(y - 6), the answer would only accept y =/= 6. Why? • I looked at the "Simplifying rational expressions" exercises and I don't see any problems where the denominator has multiple roots factored out. Note that if it asks when the simplification is valid for `(x+1) / ((x+1)(x+2))` and you simplify to `1/(x+2)`, you only need to mention that `x != -1`. Saying that `x != -2` is still there in your answer equation, so there's no need to state it again. They only want to know when the "simplification is valid", not when the whole equation is valid.
• Can somebody please tell me what I did wrong in solving this problem?
1. (x^2-36)/(6-x)
2. ((x+6)(x-6))/((-1)(x+6))
3. (x-6)/(-1)
4. (6-x), x cannot = 6 • I've seen in almost all the the equation that the domain is (All real numbers). Why can the domain not be imaginary number? and if it can be, then what happens? • You're right, there's no real reason to exclude imaginary and complex numbers. For example there is no reason the x in x^2 + 3x -5 = 0 can't be i or 5+3i.

However, we don't usually involve complex numbers without a reason. In the previous example, using real numbers gives a simple quadratic equation. If we plot it, it's just a parabola. But if you say the domain is all complex numbers, suddenly you need a 3D plot. In fact, you need 2 plots, both 3D: one for the real part and one for the imaginary part. There is no need to add that kind of complexity unless you know the input can be a complex number.

There is also a convention to use x for real variables, and z for complex ones (and n for natural).
• Is simplifying rational expressions the same as simplifying rational equations? I've been told this by a friend but I'm still not sure. • No, it is not the same, but they are related. A rational equation is a math problem that involves rational expressions. So, you would, as part of solving the rational equation, simplify the rational expressions contained within the equation. But, then you would still need to solve the rational equation after you simplified it.

So, in short, simplifying rational expressions is one of the steps involved in solving a rational equations.
• Wouldn't it be clearer to multiply the numerator and denominator both by -1 (-1/-1 = 1 after all). you would get -1(x+6)(x-6) / -1(6-x). This would turn the problem into (-x-6)(x-6) / (-6+x) and you could cancel from right there. • so then how would you solve a problem like 4/4y-8 • You can't solve it because it doesn't equal something. "Solve" is when you actually find the values of all variables, which can only happen with "equations". Note that in this video, you're dealing with "expressions". Your thing you wrote (4/4y-8) is an expression. If it was, say, 4/(4y-8) = 2, it would be an equation (note it becomes an "equation" when it "equals" something).

To simplify your expression that you wrote, you would divide both the numerator and the denominator by their common factor (4). So top divided by 4 is 1. Bottom divided by 4 is y-2. Therefore, the simplified form of 4/(4y-8) is 1/(y-2).
• As I was trying to solve the expression I factored the numerator like Sal did (x+6)(x-6) and then factored a -1 out of the denominator to set it equal to -1 (x-6) so the x-6 would cancel out, ending with the expression (x+6)/-1, and simplifying to -x-6, but I noticed instead of factoring a -1 out of the denominator, Sal multiplies the numerator with -1.

I understand that we both came to the same answer, but is Sal's method somehow more efficient or is it a matter of preference?
(1 vote) • Basically, what Sal did was to factor out (−1) from the numerator, but in the process he also chose to at least kind of explain what "factoring out" actually means, which is to multiply the expression by 1, then rewrite the 1 as a product of two factors and distribute one of those factors over the expression, thus giving us (𝑥 + 6)(𝑥 − 6) = (−1)(−1)(𝑥 + 6)(𝑥 − 6) = (−1)(𝑥 + 6)(6 − 𝑥).
• What would you do if you’re question was (x3y3)3 times xy2?  