x-2 1
___________ - ______________
x^{2 }-8x+16 x^{2}-16
x-2 1
___________ - ______________
x^{2 }-8x+16 x^{2}-16
Hello, Gaffney! My name is Vanessa and I would like to provide the written steps that Ralph took to solve the problem that you posed to us. Hopefully this will help you follow what he was able to do better.
Here are the three (3) steps:
1) Factor the denominators. Since you are subtracting rational expressions (fractions that include algebraic expressions in either the numerator or denominator) that have unlike denominators you have to create like denominators. One way to do this is by finding the least common denominator of both rational expressions (RE).
Always check to see if you can factor either (or both) denominators. Notice the denominator of the first RE is a perfect square trinomial and the denominator of the second RE is a difference of two perfect squares. (Remember that you can take the square root of the first and last terms and plug in these values into their respective forms.)
2) Create like denominators. Since the denominator of the first RE is (x - 4)^{2} or (x - 4)(x - 4) and the denominator of the second RE (x +4)(x - 4), the least common denominator would be (x - 4)(x - 4)(x +4) or (x-4)^{2} (x+4) because both denominators of the REs would have all three of those binomial expressions.
In order to make them alike we need the first RE to be multiplied by (x+4) and the second RE to be multiplied by (x - 4). (Remember to multiply BOTH the numerator and denominator with their respective binomials!)
3) Simplify. Since we are subtracting REs, and have created like denominators, we only need to subtract the numerators of both REs and let the denominators remain the same. Use the distributive property (or FOIL) to simplify the numerator of the first RE. (The numerator of the second RE will remain the same because you are simply multiplying by one.)
When you subtract a trinomial by a binomial, remember that you need to subtract BOTH terms in the binomial. In other words distribute the subtraction sign to both the x and the -4. This will result in a negative x and positive 4. (Subtracting a negative four is the same as adding a positive four.)
Combine the like terms (2x - x = 1x) and (-8 + 4 = -4) and you are left with x^{2} + x - 4. Of course, the denominator of (x - 4)^{2} (x + 4) remains the same.
I hope this helps!
Feel free to respond with any questions or comments.
Cheers,
Vanessa
(The Math Lady)
x - 2 - 1
(x - 4)^{2} (x + 4)(x - 4)
(x - 2)(x+ 4) - 1(x - 4)
(x - 4)^{2}(x + 4) (x - 4)^{2 }(x + 4)
x^{2} + 2x - 8 - x + 4
(x - 4)^{2} (x + 4)
x^{2 }+ x - 4
(x - 4)^{2} (x + 4)