Simplify completely 8x-16 divided by x2-13x+22 and find the restrictions on the variable?

When you divide the two expressions, you get  \frac{8}{x - 11}  , with the restriction that x cannot equal 2 or 11.

To find the answer and simplify, we need to start by factoring. To factor the top (numerator), we are going to look for the greatest common factor of 8x and 16. The greatest common factor would be 8.

8x - 18

8(x - 2)

Now that we have factored the numerator, we need to factor the bottom (denominator). To do this with a quadratic polynomial, we need to look for two numbers that add up to the middle term (-13) and multiply to the end term (22). The two numbers that do this are - 11 and -2. We can place them as such when we find this.

x^2 - 13x + 22

(x - 11)(x - 2)

Now that we have these factored, let's look at it expressed together.

 \frac{8(x - 2)}{(x - 11)(x - 2)}

By looking at this, we know that the restrictions are that we cannot have the denominator equal 0. Putting 2 and 11 into this equation would make the bottom 0, so we find those as our restrictions.

We will also note that the numerator and denominator both have a (x - 2), which means we can cancel it out. This leaves us with:

 \frac{8}{x - 11}

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