Rational Expression equivalent form

HINT From $\frac {3x^2}{3x-1} = \frac {\frac 13}{3x-1}+A $ you get $A = \frac {3x^2}{3x-1} - \frac {\frac 13}{3x-1}$.


You can solve the equation for A and get

$A = \frac {3x^2-\frac 13}{3x-1}$.

In the numerator 3 can be factored out.

$A = \frac {3(x^2-\frac 19)}{3x-1}$

$x^2-\frac 19$ is equivalent to the third binomial formula $a^2-b^2=(a-b)\cdot (a+b)$.

Therfore $ x^2-\frac 19=(x-\frac 13)\cdot (x+\frac 13)$

$A= \frac {3\cdot (x-\frac 13)\cdot (x+\frac 13)}{3x-1}=\frac { (3x-1)\cdot (x+\frac 13)}{3x-1}$

Numerical example: $x=2$

$\frac {3x^2}{3x-1}=\frac {3\cdot 2^2}{3\cdot 2-1}=\frac{12}{5}$

This result has to be the same like

$\frac{\frac13}{3x-1}+x+\frac 13=\frac{1}{9x-3}+x+\frac13$

$=\frac{1}{15}+2+\frac 13=\frac{1}{15}+\frac{30}{15}+\frac{5}{15}=\frac{36}{15}=\frac{12}{5}\quad \checkmark$

Tags:

Rational Numbers

Algebra Precalculus

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