Summary of "Operations on Function - Division"

Summary of “Operations on Function - Division”

This tutorial video by Senior Pablo TV focuses on solving problems related to division operations on functions, specifically rational expressions. The main goal is to find the quotient of given rational functions ( f(x) ), ( g(x) ), and ( h(x) ) in various combinations.

Main Ideas and Concepts

Division of Rational Functions: Dividing one rational function by another involves multiplying by the reciprocal of the divisor.
Factoring Polynomials: Factor both numerator and denominator of each function before division to simplify expressions.
Complex Fractions: Division often results in complex fractions, which are simplified by multiplying by the reciprocal.
Cancellation Technique: After multiplication, cancel common factors in numerator and denominator to simplify the final expression.
Restrictions on Canceling Terms: Only identical factors (binomials or monomials) can be canceled—not parts of terms or sums/differences inside parentheses.

Functions Given

[ \begin{aligned} f(x) &= \frac{x + 5}{x^2 + 2x + 1} \ g(x) &= \frac{2x^2 + 10x}{x^2 - 2x - 3} \ h(x) &= \frac{x + 5}{3x - 9} \end{aligned} ]

Factoring Details

( f(x) ):
- Numerator: ( x + 5 ) (no factorization)
- Denominator: ( x^2 + 2x + 1 = (x + 1)^2 ) (perfect square trinomial)
( g(x) ):
- Numerator: ( 2x^2 + 10x = 2x(x + 5) ) (GCF is ( 2x ))
- Denominator: ( x^2 - 2x - 3 = (x - 3)(x + 1) ) (factored trinomial)
( h(x) ):
- Numerator: ( x + 5 )
- Denominator: ( 3x - 9 = 3(x - 3) ) (GCF is 3)

Step-by-Step Methodology for Division

Rewrite the division as multiplication by the reciprocal:

[ \frac{f(x)}{g(x)} = f(x) \times \frac{1}{g(x)} = f(x) \times \text{reciprocal of } g(x) ]

Substitute the factored forms of the functions into the expression.
Multiply the numerators together and denominators together.
Cancel out common factors between numerator and denominator.
Simplify the remaining expression, optionally expanding factors if needed.

Example Solutions Presented

(a) (\frac{f(x)}{g(x)}):

Result after simplification:

[ \frac{x - 3}{2x(x + 1)} ]

Explanation: Cancelled ( x + 5 ) and ( (x + 1)^2 ) factors appropriately.

(b) (\frac{g(x)}{h(x)}):

Result after simplification:

[ \frac{6x}{x + 1} ]

Explanation: Cancelled ( x + 5 ) and ( x - 3 ), multiplied constants, and simplified.

(c) (\frac{f(x)}{h(x)}):

Viewers are encouraged to solve this on their own using the same process.

Additional Notes

The video recommends pausing to attempt problems before watching solutions.
Viewers are encouraged to review factoring techniques from earlier tutorials if needed.
Teachers are advised not to play the video directly during online classes but rather share the link.
This video is part 3 of a series on operations on functions.