Divide ((x^2-1)/(9x))÷((x^2+7x-8)/(4x))

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:

1. Multiplication by Reciprocal: To divide fractions, multiply the first fraction by the reciprocal of the second.

2. Common Factor Cancellation: When a factor appears in both the numerator and denominator, it can be canceled out.

3. Difference of Squares: This is a factoring technique used when an expression is in the form $a^2-b^2$, which factors into $(a+b)(a-b)$.

4. 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.

5. Simplification: After factoring and canceling, always simplify the expression to its lowest terms.