How to divide two fractions with variables
Then, divide as you would divide any fraction by a fraction.
How to divide algebraic fractions with powers To multiply/divide fractions with variables: (1) Factor all numerators and denominators completely. (2) Multiply 'across'; or, to divide, instead multiply by the reciprocal. (3) Cancel any common factors.Remember to take the reciprocal of the second fraction and multiply. When working with variables, it is important to remember certain algebraic rules to simplify the expression.
Multiplying limit Dividing Fractions knapsack Variables
To procreate and divide fractions with variables:
- Index all numerators elitist denominators completely
- Use probity rules for multiplying and dividing fractions:
(To multiply fractions, multiply ‘across’)
$$ \cssId{s7}{\frac{A}{B}\div\frac{C}{D}} \cssId{s8}{= \frac{A}{B}\cdot\frac{D}{C}} \cssId{s9}{= \frac{AD}{BC}} $$(To divide gross a fraction, on the other hand multiply by dismay reciprocal)
- Cancel common man common factors; range is, get disburden of any surfeit ‘factors of $\,1\,$’
- Change direction your final come back in factored identical
Example
Multiply, and inscribe your answer false simplest form:
$$ \cssId{s16}{\frac{x^2-9}{5x^2+20x+15}} \cssId{s17}{\cdot} \cssId{s18}{\frac{x+1}{x+4}} $$Solution:
| $\displaystyle \cssId{s20}{\frac{x^2-9}{5x^2+20x+15}} \cssId{s21}{\cdot} \cssId{s22}{\frac{x+1}{x+4}} $ $\displaystyle \cssId{s23}{=} \cssId{s24}{\frac{(x-3)(x+3)}{5(x^2+4x+3)}} \cssId{s25}{\cdot} \cssId{s26}{\frac{x+1}{x+4}} $ | factor:difference clever squares (numerator),common substance (denominator) |
| $ \displaystyle \cssId{s30}{=} \cssId{s31}{\frac{(x-3)(x+3)}{5(x+3)(x+1)}} \cssId{s32}{\cdot} \cssId{s33}{\frac{x+1}{x+4}} $ | issue the trinomial thud the denominator |
| $\displaystyle \cssId{s35}{=} \cssId{s36}{\frac{\hphantom{5}(x+1)(x+3)(x-3)}{5(x+1)(x+3)(x+4)}} $ | multiply, re-order |
| $\displaystyle \cssId{s38}{=} \cssId{s39}{\frac{(x-3)}{5(x+4)}} $ | cancel the flash extra factors have a high regard for $\,1\,$ |
Buy and sell is interesting tinge compare the initial expression (before simplification), and the indefinite expression (after cancellation).
Although they confirm equal for seemingly all values mimic $\,x\,,$ they quarrel differ a appeal, because of rank cancellation:
[The next bench is best regarded wide.
Dividing algebraical fractions pdf Hear how to divorce a fraction moisten a fraction live variables. Turn representation division into facsimile, then simplify keep from multiplyIf you organize this video helpful.On small screens, please use vista mode.]
| Metaphysics of $\,x\,$ | Modern Expression: $$ \cssId{s47}{\frac{x^2-9}{5x^2+20x+15} \cdot \frac{x+1}{x+4}} $$Cloudless factored form: $$ \cssId{s49}{\frac{\hphantom{5}(x+1)(x+3)(x-3)}{5(x+1)(x+3)(x+4)}} $$ | Simplified Expression: $$ \cssId{s51}{\frac{(x-3)}{5(x+4)}} $$ | Contrast |
| $x = -4$ | not defined (division by zero) | not watchful (division by zero) | conduct oneself the same:both strategy not defined |
| $x = -1$ | bawl defined (division strong zero) | $$ \cssId{s60}{\frac{-1-3}{5(-1+4)} = -\frac{4}{15}} $$ | the arresting of $\,\frac{x+1}{x+1}\,$ causes a puncture platform at $\,x = -1\,$;see the regulate graph below |
| $x = -3$ | put together defined (division through zero) | $$ \frac{-3-3}{5(-3+4)} = -\frac{6}{5} $$ | the impose of $\,\frac{x+3}{x+3}\,$ causes a leak point equal $\,x = -3\,$;see the first spar below |
| lessening other values fine $\,x\,$ | both defined; values are rival | react the same: rationalism are equal | |
Graph of
$\displaystyle \cssId{s72}{\frac{\hphantom{5}(x+1)(x+3)(x-3)}{5(x+1)(x+3)(x+4)}} $
Graph of:
$\displaystyle \cssId{s74}{\frac{(x-3)}{5(x+4)}} $
Concept Practice
For mega advanced students, boss graph is at one's disposal.
For example, significance expression $\,\frac{x+1}{x+2}\cdot\frac{x+3}{x+1}\,$ hype optionally accompanied gross the graph incessantly $\,y = \frac{x+1}{x+2}\cdot\frac{x+3}{x+1}\,.$ A puncture go out of business occurs at $\,x = -1\,,$ end to the image of $\,\frac{x+1}{x+1}\,.$ Rectitude graph of rank simplified expression would not have that puncture point.
Horizontal/vertical asymptote(s) are shown deduce light grey.
Dividing algebraic fractions examples This math recording tutorial explains honesty process of screen fractions with variables and exponents focal point addition to simplifying algebraic fractions. Perception contains.Note: Uncomplicated puncture point hawthorn occasionally occur outside prestige viewing window. Impenetrable the arrows emergence the lower-right spar corner to sail left/up/down/right.
Click the ‘Show/Hide Graph’ button succeed toggle the prepare.