Divide ((x^2-1)/(9x))÷((x^2+7x-8)/(4x))
The given problem is a question of algebraic fraction division. The aim here is to divide one rational expression by another. Specifically, the problem presents two fractions:
The numerator fraction, which is (x^2 - 1) divided by (9x).
The denominator fraction, which is (x^2 + 7x - 8) divided by (4x).
The task is to perform the division of the first fraction by the second, simplifying the expression where possible to obtain a single rational expression as the result. It involves flipping the second fraction (taking its reciprocal) to turn the division into a multiplication problem and then multiplying the numerators and denominators correspondingly. Any further simplification would typically involve factoring polynomials and canceling common factors if possible.
$\frac{x^{2} - 1}{9 x} \div \frac{x^{2} + 7 x - 8}{4 x}$
Answer
Solution:
Step 1: To divide one fraction by another, we multiply the first fraction by the reciprocal of the second. Thus, we have $\frac{x^2 - 1}{9x} \times \frac{4x}{x^2 + 7x - 8}$.
Step 2: Identify and eliminate any common factors between the numerators and denominators.
Step 2.1: Extract the factor $x$ from the denominator $9x$. This gives us $\frac{x^2 - 1}{9 \cdot x} \times \frac{4x}{x^2 + 7x - 8}$.
Step 2.2: Similarly, factor out $x$ from the numerator $4x$. We now have $\frac{x^2 - 1}{9 \cdot x} \times \frac{4 \cdot x}{x^2 + 7x - 8}$.
Step 2.3: Proceed to cancel out the common $x$ factor. The expression simplifies to $\frac{x^2 - 1}{9} \times \frac{4}{x^2 + 7x - 8}$.
Step 2.4: Rewrite the simplified expression as $\frac{x^2 - 1}{9} \times \frac{4}{x^2 + 7x - 8}$.
Step 3: Now, multiply the fractions $\frac{x^2 - 1}{9}$ and $\frac{4}{x^2 + 7x - 8}$ together. The result is $\frac{(x^2 - 1) \cdot 4}{9(x^2 + 7x - 8)}$.
Step 4: Simplify the numerator by factoring.
Step 4.1: Express $1$ as $1^2$ to prepare for factoring by the difference of squares. We get $\frac{(x^2 - 1^2) \cdot 4}{9(x^2 + 7x - 8)}$.
Step 4.2: Apply the difference of squares formula $a^2 - b^2 = (a + b)(a - b)$, where $a = x$ and $b = 1$. The numerator becomes $(x + 1)(x - 1) \cdot 4$.
Step 5: Factor the quadratic expression $x^2 + 7x - 8$ in the denominator.
Step 5.1: Look for two numbers that multiply to $-8$ (the constant term) and add up to $7$ (the coefficient of $x$). These numbers are $-1$ and $8$.
Step 5.2: Write the denominator in factored form using these numbers. The expression is now $\frac{(x + 1)(x - 1) \cdot 4}{9((x - 1)(x + 8))}$.
Step 6: Cancel out the common factor $(x - 1)$ from the numerator and denominator.
Step 6.1: Perform the cancellation to simplify the expression further to $\frac{(x + 1) \cdot 4}{9(x + 8)}$.
Step 6.2: Rewrite the simplified expression as $\frac{(x + 1) \cdot 4}{9(x + 8)}$.
Step 7: Rearrange the numerator to place the constant $4$ before the binomial $(x + 1)$. The final result is $\frac{4(x + 1)}{9(x + 8)}$.
Knowledge Notes:
Multiplication by Reciprocal: To divide fractions, multiply the first fraction by the reciprocal of the second.
Common Factor Cancellation: When a factor appears in both the numerator and denominator, it can be canceled out.
Difference of Squares: This is a factoring technique used when an expression is in the form $a^2 - b^2$, which factors into $(a + b)(a - b)$.
Factoring Quadratics: The AC method involves finding two numbers that multiply to the product of the coefficient of $x^2$ and the constant term (AC), and add up to the coefficient of $x$ (B). This helps in breaking down the middle term and factoring the quadratic expression.
Simplification: After factoring and canceling, always simplify the expression to its lowest terms.
