Divide ((x^2-16)/(x-2))÷((x^2+3x-4)/(x-8))
The problem at hand involves algebraic manipulation, specifically the division of two rational expressions—expressions that are ratios of polynomials. You are asked to divide one rational expression, ((x^2-16)/(x-2)), by another, ((x^2+3x-4)/(x-8)). This requires simplifying the complex fraction to find a single rational expression as the result. The problem will involve factoring polynomials, inverting the second rational expression to convert the problem to multiplication, and then simplifying any common factors if possible.
$\frac{x^{2} - 16}{x - 2} \div \frac{x^{2} + 3 x - 4}{x - 8}$
Answer
Solution:
Step 1:
To perform division with fractions, convert the division to multiplication by the reciprocal. Multiply $\frac{x^2 - 16}{x - 2}$ by the reciprocal of $\frac{x^2 + 3x - 4}{x - 8}$, which is $\frac{x - 8}{x^2 + 3x - 4}$.
Step 2:
Begin by simplifying the first fraction's numerator.
Step 2.1:
Express $16$ as $4^2$ to facilitate factoring. $\frac{x^2 - 4^2}{x - 2} \cdot \frac{x - 8}{x^2 + 3x - 4}$
Step 2.2:
Apply the difference of squares formula, $a^2 - b^2 = (a + b)(a - b)$, where $a = x$ and $b = 4$. $\frac{(x + 4)(x - 4)}{x - 2} \cdot \frac{x - 8}{x^2 + 3x - 4}$
Step 3:
Factor the expression $x^2 + 3x - 4$ by searching for two numbers that multiply to $-4$ (the constant term) and add up to $3$ (the coefficient of the linear term).
Step 3.1:
Identify the numbers as $-1$ and $4$.
Step 3.2:
Rewrite the factored form using these integers. $\frac{(x + 4)(x - 4)}{x - 2} \cdot \frac{x - 8}{(x - 1)(x + 4)}$
Step 4:
Eliminate the common factors present in both the numerator and the denominator.
Step 4.1:
Isolate the common factor of $x + 4$ from $(x - 1)(x + 4)$. $\frac{(x + 4)(x - 4)}{x - 2} \cdot \frac{x - 8}{(x + 4)(x - 1)}$
Step 4.2:
Cross out the common factor of $x + 4$. $\frac{(\cancel{x + 4})(x - 4)}{x - 2} \cdot \frac{x - 8}{(\cancel{x + 4})(x - 1)}$
Step 4.3:
Simplify the expression after cancellation. $\frac{x - 4}{x - 2} \cdot \frac{x - 8}{x - 1}$
Step 5:
Finally, multiply $\frac{x - 4}{x - 2}$ by $\frac{x - 8}{x - 1}$. $\frac{(x - 4)(x - 8)}{(x - 2)(x - 1)}$
Knowledge Notes:
Division of Fractions: To divide one fraction by another, multiply the first fraction by the reciprocal of the second.
Difference of Squares: This is a special factoring technique where $a^2 - b^2$ can be factored into $(a + b)(a - b)$.
Factoring Quadratic Equations: The AC method involves finding two numbers that multiply to the product of the quadratic's leading coefficient and constant term (AC), and add up to the middle term's coefficient (B). Then, the quadratic can be factored into binomials.
Cancellation: When a factor appears in both the numerator and denominator of a fraction, it can be cancelled out, simplifying the expression.
Multiplication of Fractions: To multiply fractions, multiply the numerators together to get the new numerator, and multiply the denominators together to get the new denominator.
