Simplifying and Factoring Rational Expressions

The Golden Rule of Division: When dividing rational expressions, the first expression (the dividend) is never flipped. Only the expression being divided by (the divisor) is flipped to its reciprocal.
General Strategy: To simplify these expressions, one must perform four distinct factoring steps—factoring the numerator and denominator of both the first expression and the flipped second expression.
Step-by-Step Factoring of a Sample Expression: * Factoring the First Numerator: In the example $2x^3 - 12x^2$ , the greatest common factor is $2x^2$ . * The constant $2$ goes into both terms ( $2$ and $12$ ). * Between $x^3$ and $x^2$ , the smaller power ( $x^2$ ) is pulled out. * Dividing $2x^3$ by $2x^2$ yields $x$ . * Dividing $-12x^2$ by $2x^2$ yields $-6$ . * Factored result: $2x^2(x - 6)$ . * Factoring the First Denominator ("Double Bubble"): For the quadratic $x^2 - 4x - 12$ , search for two numbers that multiply to get $-12$ and add/subtract to get $-4$ . * Since the last term is negative, the signs in the bubbles must be a plus and a minus: $(x + \text{?})(x - \text{?})$ . * The factors of $12$ are $2$ and $6$ . Because the middle term is $-4$ , the larger factor ( $6$ ) must be paired with the negative sign. * Factored result: $(x + 2)(x - 6)$ .

Sign Conventions in Factoring: * If the constant term (the last number) is positive, both signs in the binomial factors must be the same (either both positive or both negative). * To determine which sign to use, look at the middle term. If the middle term is negative (e.g., $-8x$ ), the signs are minus-minus: $(x - \text{?})(x - \text{?})$ . * If the middle term were positive (e.g., $+8x$ ), the signs would be plus-plus: $(x + \text{?})(x + \text{?})$ .
Validating Factors via Arithmetic: Always perform a quick check. For the expression $x^2 - 8x + 15$ , the factors are $-5$ and $-3$ . * Check: $-5 \times -3 = 15$ and $-5 + (-3) = -8$ . If the arithmetic holds, the signs and numbers are correct.
Factoring with Common Monomials: In the expression $24x^2 - 8x^3$ (flipped during division logic): * Identify the greatest common constant: $8$ goes into both $24$ and $8$ . * Identify the smallest variable power: $x^2$ . * Factoring out $8x^2$ yields: $8x^2(3 - x)$ .

Recognizing Opposite Binomials: When terms in a bubble are identical but have reversed signs (e.g., $(x - 3)$ and $(3 - x)$ ), they do not cancel to $1$ . Instead, they cancel to $-1$ .
The Substantiation Proof: To demonstrate why this works for any number, consider a variable substituted with a specific value (e.g., the "football number" $66$ ). * Case 1: $(66 - 3) = 63$ * Case 2: $(3 - 66) = -63$ * The ratio of $63 / -63$ is $-1$ . This rule remains constant regardless of the value substituted.
Placement of the Negative Sign: When cancellation results in a $-1$ , the negative sign can be placed with the numerator, the denominator, or out in front of the entire fraction. It is purely a matter of preference, though consistency is recommended.

Initial Setup: High-level rational expressions often require handling multiple factors simultaneously.
Simplifying Constants and Monomials: * If you have a $2x^2$ in the numerator and an $8x^2$ in the denominator, the $x^2$ terms cancel out completely because they are part of a multiplication-only relationship (no addition signs between them and the coefficients). * The fraction $2/8$ simplifies to $1/4$ using "old school fraction" reduction rules.
Constructing the Final Expression: * Numerator: After cancellation, if a $(x - 5)$ bubble and a $-1$ remain, the numerator becomes $-1(x - 5)$ . * Denominator: If a $4$ and a $(x + 2)$ bubble remain, the denominator becomes $4(x + 2)$ . * The final answer can be written as $\frac{-1(x - 5)}{4(x + 2)}$ . Distributing the terms is possible but not necessarily required unless solving further for algebra 3 or college algebra contexts.

Reciprocals of Non-Fractional Expressions: When dividing by an expression that is not in fraction form, such as $(4x + 1)$ , its reciprocal is $\frac{1}{4x + 1}$ .
Handling Mixed Operations: In a string of operations involving both division and multiplication, only the terms following a division sign are flipped. Terms following a multiplication sign remain as they are.
Difference of Perfect Squares: Recognizing patterns is essential for quick factoring. * Example: $16x^2 - 1$ is a difference of squares. * Criteria: There must be two terms, a minus sign, and both terms must be perfect squares. * Factoring Process: Find the square root of each term. $\sqrt{16x^2} = 4x$ and $\sqrt{1} = 1$ . * Result: $(4x + 1)(4x - 1)$ .
Final Cancellation Example: * Expression: $\frac{x + 3}{x} \times \frac{1}{4x + 1} \times \frac{(4x + 1)(4x - 1)}{x + 3}$ . * The $(x + 3)$ in the numerator and denominator cancel out. * The $(4x + 1)$ in the denominator and numerator cancel out. * Result: Only $\frac{4x - 1}{x}$ remains.

Question on Flipping: A student asked if the first expression should be flipped. The instructor clarified that you always flip the one you are dividing by, never the first one.
Discussion on Football Numbers: The instructor asked Dagore for his football number to use as a substitution example. Dagore provided the number $66$ . The instructor used this to show that $(66 - 3) / (3 - 66)$ equals $-1$ , proving the rule for cancelling binomials with reversed signs.
Question on Placing the Negative: A student asked whether the $-1$ should be placed in the top or bottom of the fraction. The instructor responded that it doesn't matter as long as the entire expression reflects the negative value, though the student picked the "top" as the preferred placement for the example.