A common denominator is a denominator that can be used to rewrite rational expressions with different denominators, so that they have the same denominators.
We can always rewrite a rational expression by multiplying its numerator and denominator by the samepolynomial.
1/x^2=1* 2/x^2* 2 and 3/2x=3* x/2x* x
Therefore, given two rational expressions, we can always multiply the numerator and the denominator of one expression by the denominator of the other one.
Rational Expressions
Rewrite Expression 1
Rewrite Expression 2
2/x^2 and 1/2x
2* 2x/x^2* 2x
1* x^2/2x* x^2
1/x and 2/x-1
1*( x-1)/x*( x-1)
2* x/(x-1)* x
If we rewrite the first expression using the denominator of the second expression and the second expression using the denominator of the first expression, the denominators of the obtained expressions consist of exactly the same factors — they are identical. Therefore, your friend is correct and her method works.
b
Let's take a look at two pairs of rational expressions. We will find their least common denominator (LCD) and their common denominator using your friend's method. Then, we will compare the denominators obtained using these two methods.
Rational Expressions
Factor Denominators
LCD
Your Friend's Method
Are the Results the Same?
1/2x, 1/x^2
x= 2* x and x^2= x* x
2* x* x=2x^2
2x* x^2=2x^3
No *
1/x-1, 1/x+1
x-1 and x+1
( x-1)( x+1)=x^2-1
(x-1)(x+1)=x^2-1
Yes ✓
We can see that the method of finding the product of the denominators, does not always give us the least common denominator. To explain why, we will consider two cases.
The denominators have no common factors.
The denominators have common factors.
Case 1
Take a look at examples when the denominators of rational expressions have no common factors.
Rational Expressions
LCD
Your Friend's Method
2/y+2, y/y-1
(y+2)(y-1)=y^2+y-2
(y+2)(y-1)=y^2+y-2
z+1/z^2, 5/z-2
z^2(z-2)=z^3-2z^2
z^2(z-2)=z^3-2z^2
If the denominators have no common factors, their product is the LCD. Therefore, in these cases, your friend's method is equivalent to finding the LCD.
Case 2
Let's consider examples when the denominators of rational expressions have common factors.
Rational Expressions
Factor Denominators
LCD
Your Friend's Method
2/y^2-y, y/y-1
y^2-y= y( y-1) and y-1
y( y-1)=y^2-y
(y^2-y)(y-1)=y^3-2y^2+y
z+1/2z^2, 5/4z
2z^2= 2* z* z and 4z= 2* 2* z
2* 2* z* z=4z^2
2z^2* 4z=8z^3
If the denominators have common factors, the LCD is the product of their prime factors, each raised to the greatest power that occurs in the expressions. Therefore, using your friend's method causes us to repeat some of the factors more times than necessary and we do not obtain the LCD.
