Compare numerical fractions against rational expressions. What is common and what is different about them, and how do we make sure that a rational expression does not get undefined?

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embizze | High School Teacher | (Level 1) Educator Emeritus

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Compare fractions to rational expressions:

(1) To simplify either you factor and then cancel any factor that appears in numerator and denominator.

`21/15=(3*7)/(3*5)=7/5`

`(x^2+2x+1)/(x^2+4x+3)=((x+1)^2)/((x+1)(x+3))=(x+1)/(x+3)`

Note that `21/15=7/5` ; on the other hand `(x^2+2x+1)/(x^2+4x+3)=(x+1)/(x+3)` for all x except x=-1, thus the two expressions are not equivalent.

(2) To add/subtract you get a common denominator then add/subtract the numerators:

`2/3+3/4=8/12+9/12=17/12`

`2/(x+3)+3/(x+1)=(2(x+1))/((x+3)(x+1))+(3(x+3))/((x+1)(x+3))=(5x+11)/((x+1)(x+3))`

(3) To multiply you multiply numerators and denominators:

`2/3*3/4=(2*3)/(3*4)=6/12=1/2`

`1/(x+1)*3/x=(1*3)/(x(x+1))=3/(x^2+x)`

(4) Both will have additive inverses found by multiplying by -1

(5) Both will have multiplicative inverses (it will be the reciprocal) except when they are 0.

`2/3*3/2=1` but `0/7` does not have a multiplicative inverse.

The multiplicative inverse of `(x+1)/(x+3)` is `(x+3)/(x+1)` except when x=-1

(6) Numerical fractions exist except when the denominator is zero.

Rational expressions are defined as long as the denominator is not zero.

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