Divide Rational Expressions

Dividing Rational Expressions

When dividing rational expressions, the "Keep Change Flip" method is the foundational technique used to transform a division problem into a multiplication problem, which is generally easier to handle [1].

Keep Change Flip Method Steps:

1. Keep the first fraction exactly as it is without any alterations [1].

2. Change the division operation (the division symbol \/) into a multiplication operation (the multiplication symbol $*$) [1].

3. Flip the second fraction by taking its reciprocal. This means the numerator of the second fraction becomes its new denominator, and the denominator becomes its new numerator [1].

4. After successfully applying the Keep Change Flip method, the expression is now a multiplication of rational expressions. The next crucial step is to factor all numerators and all denominators completely into their simplest irreducible forms. This includes looking for common factors, difference of squares, sum/difference of cubes, and trinomial factorizations [1].

5. Once all expressions are factored, it's essential to determine excluded values. These are any values of the variable that would make any denominator zero at any point during the division process, including the denominator of the original second fraction before it was flipped, as well as all denominators in the final multiplied expression. These values must be excluded because division by zero is undefined [1].

6. Cancel any common factors that appear in both the numerator and the denominator across the entire expression. This simplification step is only possible after complete factorization [1].

Example 1: Divide (x^2 - 9) \/ (4x + 20) by $(x^2 + 8x + 15)$ [1]

• Step 1: Rewrite as a fraction (if necessary) and apply Keep Change Flip.

  • The problem is initially presented as (x^2 - 9) \/ (4x + 20) divided by $(x^2 + 8x + 15)$. To make KCF clearer, envision the second term as a fraction: (x^2 + 8x + 15) \/ 1 [1].

  • The original expression can be more clearly thought of as \left[(x^2 - 9) \/ (4x + 20)\right] divided by \left[(x^2 + 8x + 15) \/ 1\right] [1].

  • Correction/Clarification: The prompt implies the division is (x^2 - 9) \/ (4x + 20) divided by just $(x^2 + 8x + 15)$. This is often interpreted as $\frac{\frac{x^2-9}{4x+20}}{x^2+8x+15}$. If it's $(x^2 - 9) / (4x + 20) \div (x^2 + 8x + 15)$, where $(x^2 + 8x + 15)$ is the entire second term (which has an implicit denominator of 1), then the KCF application is different from the original note's interpretation of $(x^2 - 9)$ being divided by $(4x+20)$ which then is divided by $(x^2+8x+15)$. Let's stick to the interpretation that matches the provided solution: $\frac{x^2 - 9}{1} \div \frac{4x + 20}{x^2 + 8x + 15}$ This requires the problem statement to be $(x^2 - 9)$ divided by the rational expression $(4x + 20) / (x^2 + 8x + 15)$. Assuming this revised interpretation for consistency with the note's solution:

    • Keep: $(x^2 - 9)$ (or (x^2 - 9) \/ 1) [1].

    • Change: division to multiplication ($\times$) [1].

    • Flip: (x^2 + 8x + 15) \/ (4x + 20) [1].

  • Resulting expression after KCF: $\frac{(x^2 - 9)}{1} \times \frac{(x^2 + 8x + 15)}{(4x + 20)}$ [1].

• Step 2: Factor all expressions completely.

  • $(x^2 - 9)$: This is a classic difference of squares pattern ($a^2 - b^2 = (a+b)(a-b)$). Here, $a=x$ and $b=3$. Factors to $(x + 3)(x - 3)$ [1].

  • $(x^2 + 8x + 15)$: This is a quadratic trinomial ($ax^2 + bx + c$). We need to find two numbers that multiply to $c$ (15) and add to $b$ (8). These numbers are 3 and 5. Factors to $(x + 3)(x + 5)$ [1].

  • $(4x + 20)$: Find the Greatest Common Factor (GCF) between 4x and 20, which is 4. Factors to $4(x + 5)$ [1].

• Step 3: Determine Excluded Values.

  • Excluded values are any values of $x$ that would make a denominator zero at any point in the original problem or in the intermediate steps before cancellation. This includes the denominator of the original second fraction before it was flipped, as well as all denominators in the factored expression [1].

  • From the original denominator of the second fraction (which was $4x + 20$ before flipping): Set $4x + 20 = 0 \Rightarrow 4x = -20 \Rightarrow x = \mathbf{-5}$ [1].

  • From the new denominator ($4x+20$ after flipping). (This is the same as the previous point)

  • From the original implied denominator of the second term before it was flipped (if it was part of a larger rational expression). In this specific problem's interpretation, the second term was initially $(4x+20)/(x^2+8x+15)$ meaning the $x^2+8x+15$ must NOT be zero. So we consider factors from this original denominator as well: Set $(x^2 + 8x + 15) = 0 \Rightarrow (x + 3)(x + 5) = 0 \Rightarrow x = \mathbf{-3}$ or $x = \mathbf{-5}$ [1].

  • Therefore, the excluded values are $-5$ and $-3$ [1].

• Step 4: Cancel common factors.

  • Substitute the factored forms back into the expression: $\frac{(x + 3)(x - 3)}{1} \times \frac{(x + 3)(x + 5)}{4(x + 5)}$ [1].

  • Identify common factors in the numerator and denominator. We can cancel the factor $(x + 5)$ from the numerator of the second fraction and the denominator of the second fraction [1].

  • The remaining factors are $(x + 3)$, $(x - 3)$, and another $(x + 3)$ in the numerator, and $4$ in the denominator [1].

• Step 5: Write the final answer.

  • Combine the remaining factors by multiplying across: $\frac{(x + 3)(x - 3)(x + 3)}{4}$ [1].

  • Simplify by combining like terms: $\mathbf{\frac{(x + 3)^2 (x - 3)}{4}}$ [1].

Example 2: Simplify a complex rational expression $(4x^3 + 32) / (6x^2 - 24) / (3x^2 - 6x + 12) / (x^2 - 7x + 10)$ [1]

• Step 1: Rewrite the complex fraction as a division problem.

  • The given format implies the first rational expression is being divided by the second rational expression: $\left[\frac{(4x^3 + 32)}{(6x^2 - 24)}\right]$ divided by $\left[\frac{(3x^2 - 6x + 12)}{(x^2 - 7x + 10)}\right]$ [1].

• Step 2: Apply Keep Change Flip.

  • Keep: The first fraction $\frac{(4x^3 + 32)}{(6x^2 - 24)}$ remains unchanged [1].

  • Change: The division sign becomes a multiplication sign [1].

  • Flip: The second fraction is inverted: $\frac{(x^2 - 7x + 10)}{(3x^2 - 6x + 12)}$ [1].

  • Resulting expression: $\left[\frac{(4x^3 + 32)}{(6x^2 - 24)}\right] \times \left[\frac{(x^2 - 7x + 10)}{(3x^2 - 6x + 12)}\right]$ [1].

• Step 3: Factor all expressions completely.

  • $(4x^3 + 32)$:

    • First, take out the GCF, which is 4: $4(x^3 + 8)$ [1].

    • Recognize $(x^3 + 8)$ as a sum of perfect cubes ($a^3 + b^3 = (a + b)(a^2 - ab + b^2)$). Here, $a = x$ and $b = 2$ (since $x^3 = x^3$ and $8 = 2^3$) [1].

    • Factor the sum of cubes: $(x + 2)(x^2 - 2x + 4)$ [1].

    • The full factorization is $4(x + 2)(x^2 - 2x + 4)$ [1]. (Note: The quadratic factor $(x^2 - 2x + 4)$ cannot be factored further over real numbers) [1].

  • $(6x^2 - 24)$:

    • First, take out the GCF, which is 6: $6(x^2 - 4)$ [1].

    • Recognize $(x^2 - 4)$ as a difference of squares ($a^2 - b^2 = (a+b)(a-b)$). Factors to $(x + 2)(x - 2)$ [1].

    • The full factorization is $6(x + 2)(x - 2)$ [1].

  • $(x^2 - 7x + 10)$:

    • This is a quadratic trinomial. Find two numbers that multiply to 10 and add to -7. These numbers are -5 and -2. Factors to $(x - 5)(x - 2)$ [1].

  • $(3x^2 - 6x + 12)$:

    • First, take out the GCF, which is 3: $3(x^2 - 2x + 4)$ [1]. (Note: The quadratic factor $(x^2 - 2x + 4)$ cannot be factored further over real numbers, similar to the one above) [1].

• Step 4: Cancel common factors and simplify constants.

  • Substitute all factored forms back into the expression:
    $\frac{4(x + 2)(x^2 - 2x + 4)}{6(x + 2)(x - 2)} \times \frac{(x - 5)(x - 2)}{3(x^2 - 2x + 4)}$ [1].

  • Identify and cancel common factors present in both numerators and denominators:

    • Cancel $(x + 2)$ [1].

    • Cancel $(x - 2)$ [1].

    • Cancel $(x^2 - 2x + 4)$ [1].

  • Now, simplify the numerical coefficients: The $4$ from the first numerator and $6$ from the first denominator simplify. 4 \/ 6 reduces to 2 \/ 3 [1].

  • The remaining terms are $2$ and $(x - 5)$ in the numerator, and $3$ and $3$ in the denominator [1].

• Step 5: Write the final answer.

  • Combine the remaining numerators and denominators: $\frac{2 \times (x - 5)}{3 \times 3}$ [1].

  • Simplify to: $\mathbf{\frac{2(x - 5)}{9}}$ [1].

Dividing Polynomials

A polynomial is an algebraic expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables (e.g., $2x^3 - 5x + 7$) [1].

Dividing a Polynomial by a Monomial

When dividing a polynomial by a monomial (an expression with a single term, like $4x$ or $9x^2$), the most straightforward approach is to separate the fraction into multiple fractions. Each individual term of the polynomial (the numerator) is divided by the monomial (the denominator) [1].

Example 1: Divide $(8x^3 - 4x^2 + 12x)$ by $4x$ [1]

• Step 1: Separate into individual terms divided by the monomial.

  • The expression becomes the sum/difference of three separate fractions: $\frac{8x^3}{4x} - \frac{4x^2}{4x} + \frac{12x}{4x}$ [1].

• Step 2: Divide each term.

  • For \mathbf{8x^3 \/ 4x}:

    • Divide the coefficients: 8 \/ 4 = 2 [1].

    • Divide the variables using the exponent rule x^m \/ x^n = x^{m-n}: x^3 \/ x^1 = x^{(3-1)} = x^2 [1].

    • Result for this term: $2x^2$ [1].

  • For \mathbf{-4x^2 \/ 4x}:

    • Divide the coefficients: -4 \/ 4 = -1 [1].

    • Divide the variables: x^2 \/ x^1 = x^{(2-1)} = x^1 = x [1].

    • Result for this term: $-x$ [1].

  • For \mathbf{12x \/ 4x}:

    • Divide the coefficients: 12 \/ 4 = 3 [1].

    • Divide the variables: x^1 \/ x^1 = x^{(1-1)} = x^0 = 1 (any non-zero number raised to the power of 0 is 1) [1].

    • Result for this term: $3$ [1].

• Step 3: Combine the results.

  • Add/subtract the results of each term: $2x^2 - x + 3$ [1].

Example 2: Divide $(36x^5 - 63x^4 + 72x^3)$ by $9x^2$ [1]

• Step 1: Separate into individual terms divided by the monomial.

  • Set up as: $\frac{36x^5}{9x^2} - \frac{63x^4}{9x^2} + \frac{72x^3}{9x^2}$ [1].

• Step 2: Divide each term.

  • For \mathbf{36x^5 \/ 9x^2}:

    • 36 \/ 9 = 4 [1].

    • x^5 \/ x^2 = x^{(5-2)} = x^3 [1].

    • Result for this term: $4x^3$ [1].

  • For \mathbf{-63x^4 \/ 9x^2}:

    • -63 \/ 9 = -7 [1].

    • x^4 \/ x^2 = x^{(4-2)} = x^2 [1].

    • Result for this term: $-7x^2$ [1].

  • For \mathbf{72x^3 \/ 9x^2}:

    • 72 \/ 9 = 8 [1].

    • x^3 \/ x^2 = x^{(3-2)} = x^1 = x [1].

    • Result for this term: $8x$ [1].

• Step 3: Combine the results.

  • Combine the simplified terms: $4x^3 - 7x^2 + 8x$ [1].

Dividing a Polynomial by a Binomial (Long Division)

When the denominator (divisor) has two or more terms (e.g., a binomial like $2x - 3$ or a trinomial), you must use polynomial long division [1]. This method is analogous to numerical long division.

General Process for Polynomial Long Division:

1. Set up the long division in the standard format. The numerator (dividend) goes inside the division symbol, and the denominator (divisor) goes outside [1]. Crucial step: If there are any missing terms in the dividend (e.g., no $x^2$ term in a cubic polynomial), insert placeholders with a coefficient of 0 (e.g., $0x^2$). This ensures proper alignment during subtraction [1].

2. Divide: Focus only on the first term of the dividend and the first term of the divisor. Divide the first term of the dividend by the first term of the divisor. This result is the first term of your quotient. Write this term directly above the corresponding term in the dividend [1].

3. Multiply: Take the term you just wrote in the quotient and multiply it by the entire divisor. Write this product directly below the corresponding terms of the dividend. Ensure like terms are aligned vertically [1].

4. Subtract: Draw a line below the product, and change the signs of all terms in the product you just wrote. Then, combine these terms with the corresponding terms above them in the dividend. This step is critical; a common error is forgetting to change signs. The first term should always cancel out to zero [1].

5. Bring down: Bring down the next unused term from the original dividend to the working line [1].