Simplify each expression.
1. 2x+8/x^2-16
2. 1+x/x^2+x
3. x^2-12x-28/x^2-14x
4. x^2+7x+10/x^2-4
You probably meant
1. (2x+8)/(x^2-16)
then this factors to
2(x+4)/[(x+4)(x-4)]
= 2/(x-4), where x is not equal to -4
With the proper use of brackets, the others factor as well.
Let me know how you made out with them.
Thank you so much.
1. (x^2-12x-28)/(x^2-14x)
Answer: (x+2)/(x)
2. (x^2+7x+10)/(x^2-4)
Answer: (x+5)/(x-2)
3. (y^2-16)/(y^2-7y+12)
Answer: (y+4)/(y-3)
Could you check my answers for these questions. I will really appeciate it. Thank you so much again.
Your answers are all correct.
Have you learned about restrictions?
e.g. in the first one you probably had
(x-14)(x+2)/[(x(x-14)] , then canceled the x-14.
but suppose x = 14. Didn't you just divide by zero ?
That is why we have to exclude the value of the variable which would have made that denominator zero.
so the complete answer to that one would have been ...
(x+2)/x , x not equal to 14
No, I haven't. Thank you so much.
To simplify each expression, we need to factorize the denominators where possible and simplify any common factors in the numerator and denominator. Let's go through each expression step by step:
1. For the expression 2x + (8 / (x^2 - 16)), we first factorize the denominator (x^2 - 16):
x^2 - 16 = (x - 4)(x + 4)
Now, we can rewrite the expression as follows:
2x + (8 / (x^2 - 16)) = 2x + (8 / ((x - 4)(x + 4)))
Since there are no common factors between the numerator and the denominator, this expression cannot be simplified any further.
2. For the expression 1 + (x / (x^2 + x)), there aren't any common factors that can be simplified.
3. For the expression (x^2 - 12x - 28) / (x^2 - 14x), we can factorize the numerator and denominator:
(x^2 - 12x - 28) = (x - 14)(x + 2)
(x^2 - 14x) = x(x - 14)
Now we can rewrite the expression as:
(x^2 - 12x - 28) / (x^2 - 14x) = ((x - 14)(x + 2)) / (x(x - 14))
Notice that we have a common factor of (x - 14) in both the numerator and the denominator. We can simplify the expression by canceling out this common factor:
(x^2 - 12x - 28) / (x^2 - 14x) = (x + 2) / x
4. For the expression (x^2 + 7x + 10) / (x^2 - 4), we can factorize the numerator and denominator:
(x^2 + 7x + 10) = (x + 5)(x + 2)
(x^2 - 4) = (x - 2)(x + 2)
Now we can rewrite the expression as:
(x^2 + 7x + 10) / (x^2 - 4) = ((x + 5)(x + 2)) / ((x - 2)(x + 2))
Notice that we have a common factor of (x + 2) in both the numerator and the denominator. We can simplify the expression by canceling out this common factor:
(x^2 + 7x + 10) / (x^2 - 4) = (x + 5) / (x - 2)
By following these steps, we simplified each given expression as much as possible.