Multiplication of Rational Functions

Remember that a rational function is one represented as a fraction. Multiply rational functions by following the usual procedure for multiplying any fraction:

Step 1) Multiply the numerator by the other numerator using FOIL Method.

Step 2) Multiply the denominator by the other denominator using FOIL Method.

Step 3) Reduce/simplify the fraction (if needed)

Example A:

\frac{x + 1}{x + 3} * \frac{2 x + 3}{x - 1}

$\frac{x+1}{x+3}*\frac{2x+3}{x-1}$

Start by multiplying the numerators first:

(x + 1) (2 x + 3) = 2 x^{2} + 5 x + 3

$(x+1)(2x+3)=2x^2+5x+3$

Now multiply the denominators:

(x + 3) (x - 1) = x^{2} + 2 x - 3

$(x+3)(x-1) = x^2+2x-3$

Final answer:

\frac{2 x^{2} + 5 x + 3}{x^{2} + 2 x - 3}

$\frac{2x^2 + 5x + 3}{x^2 + 2x - 3}$

Example B:

\frac{x^{2} - 4}{x - 3} * \frac{x^{2} - 7 x + 12}{x^{2} - 2 x}

$\frac{x^2 - 4}{x - 3}*\frac{x^2 - 7x + 12}{x^2 - 2x}$

Step 1) Multiply numerators:

(x^{2} - 4) (x^{2} - 7 x + 12) = x^{4} - 7 x^{3} + 8 x^{2} - 28 x - 48

$(x^2-4)(x^2-7x+12) = x^4-7x^3+8x^2-28x-48$

2) Cancel where you can.

We can cancel the following: (x - 3) with (x - 3) and (x - 2) with (x - 2).

After doing so, we are left with the final answer: (x + 2) ( x - 4)/x

You can leave the final answer as shown above (called factored form) or you can write your final answer in standard form by multiplying to simplify.

Look at the difference:

Factored Form For Sample B

(x + 2) ( x - 4)/x

Standard Form For Sample B

x^2 - 2x - 8/x

By Mr. Feliz
(c) 2005