Write each rational expression in simplest form and list the values of the variables for which the fraction is undefined.
1. x^2-7x+12/x^2+2x-15
2. 5y^2-20/y^2+4y+4
3. -7+7a/21a^2-21
4. 5(1-b)+15/b^2-16
#2:
5(y^2-4)/(y+2)^2
5(y+2)(y-2)/(y+2)^2
5(y-2)/(y+2)
undefined where y+2=0, or y = -2.
Do the others in like wise. What do you get?
To simplify these rational expressions and find the values of the variables for which the expressions are undefined, we can follow these steps:
1. Factor both the numerator and the denominator, if possible.
2. Cancel out any common factors.
3. Identify the values of the variables for which the denominator is equal to zero, as these values would make the expression undefined.
Let's go through each expression step by step:
1. x^2-7x+12/x^2+2x-15
First, factor the numerator and denominator:
Numerator: (x-3)(x-4)
Denominator: (x+5)(x-3)
Now, cancel out the common factor of (x-3):
(x-4)/(x+5)
The fraction is undefined when the denominator, (x+5), is equal to zero. Therefore, x cannot equal -5.
2. 5y^2-20/y^2+4y+4
The numerator is a multiple of 5, so we can simplify by factoring out 5:
5(y^2-4)/(y^2+4y+4)
Next, factor the numerator and denominator:
Numerator: 5(y+2)(y-2)
Denominator: (y+2)(y+2)
Cancel out the common factor of (y+2):
5(y-2)/(y+2)
The fraction is undefined when the denominator, (y+2), is equal to zero. Therefore, y cannot equal -2.
3. -7+7a/21a^2-21
First, factor out a common factor of 7:
7(a - 1)/(21a^2 - 21)
Next, factor the denominator:
21(a^2 - 1)
Apply the difference of squares to further simplify the denominator:
21(a + 1)(a - 1)
Cancel out the common factor of (a - 1):
7(a + 1)/21(a + 1)
Simplifying further, we get:
(a + 1)/3
The fraction does not have any values for which it is undefined.
4. 5(1-b)+15/b^2-16
First, simplify the numerator:
5 - 5b + 15/b^2 - 16
Combining like terms in the numerator, we get:
-5b - 11 + 15/b^2 - 16
Simplifying further, we get:
-5b - 27 + 15/b^2
The fraction is undefined when the denominator, (b^2 - 16), is equal to zero. To find these values, we factor the denominator:
(b + 4)(b - 4)
Therefore, b cannot equal 4 or -4.
To simplify a rational expression, we need to factor both the numerator and denominator (if possible) and then cancel out any common factors between them.
Let's work through each expression step by step:
1. x^2-7x+12/x^2+2x-15
Factor the numerator: (x-3)(x-4)
Factor the denominator: (x+5)(x-3)
We can cancel out the common factor of (x-3) between the numerator and denominator:
(x-3)(x-4)/(x+5)(x-3)
Simplified form: (x-4)/(x+5)
To find the values of x for which the fraction is undefined, we need to identify the values that would make the denominator equal to zero. In this case, x cannot be -5 because it would make the denominator (x+5) equal to zero, resulting in division by zero.
Therefore, the fraction is undefined for x=-5.
2. 5y^2-20/y^2+4y+4
Factor the numerator: 5(y^2-4)
Factor the denominator: (y+2)(y+2)
We can cancel out the common factor of (y+2) between the numerator and denominator:
5(y^2-4)/(y+2)(y+2)
Simplified form: 5(y-2)/(y+2)
To find the values of y for which the fraction is undefined, we need to identify the values that would make the denominator equal to zero. In this case, y cannot be -2 because it would make the denominator (y+2) equal to zero, resulting in division by zero.
Therefore, the fraction is undefined for y=-2.
3. -7+7a/21a^2-21
There is no common factor to cancel out between the numerator and denominator.
We leave it as it is:
-7+7a/(21a^2-21)
To find the values of a for which the fraction is undefined, we need to identify the values that would make the denominator equal to zero. In this case, a cannot be 0 because it would make the denominator (21a^2-21) equal to zero, resulting in division by zero.
Therefore, the fraction is undefined for a=0.
4. 5(1-b)+15/b^2-16
Simplify the numerator: 5-5b+15
Simplify the denominator: (b+4)(b-4)
Combine like terms in the numerator:
20-5b
The expression becomes:
(20-5b)/(b+4)(b-4)
To find the values of b for which the fraction is undefined, we need to identify the values that would make the denominator equal to zero. In this case, b cannot be -4 and 4 because they would make the denominator (b+4)(b-4) equal to zero, resulting in division by zero.
Therefore, the fraction is undefined for b=-4 and b=4.