Simplify (x+2+3/(x-4))/(x-3-2/(x-4))
The question asks you to simplify a complex rational expression. The expression is composed of a numerator and a denominator, both of which include polynomials and a fraction nested within them. The goal is to perform algebraic manipulations such that the expression is rewritten in a more straightforward, reduced form, ideally without complex fractions or unnecessary terms. This typically involves common algebraic techniques such as combining like terms, multiplying or dividing both the numerator and denominator by common factors to simplify the complex fraction, and perhaps factoring polynomials if that aids in the simplification process.
$\frac{x + 2 + \frac{3}{x - 4}}{x - 3 - \frac{2}{x - 4}}$
Answer
Solution:
Step 1
Scale both the top and bottom of the fraction by $(x - 4)$.
Step 1.1
Multiply $\frac{x + 2 + \frac{3}{x - 4}}{x - 3 - \frac{2}{x - 4}}$ by $\frac{x - 4}{x - 4}$ to get $\frac{(x - 4)(x + 2 + \frac{3}{x - 4})}{(x - 4)(x - 3 - \frac{2}{x - 4})}$.
Step 1.2
Combine terms to form $\frac{(x - 4)(x + 2 + \frac{3}{x - 4})}{(x - 4)(x - 3 - \frac{2}{x - 4})}$.
Step 2
Distribute $(x - 4)$ across the terms in both the numerator and denominator.
Step 3
Eliminate common factors.
Step 3.1
Remove the $(x - 4)$ term where possible.
Step 3.1.1
Cancel out $(x - 4)$ to simplify $\frac{(x - 4)x + (x - 4) \cdot 2 + 3}{(x - 4)x - (x - 4) \cdot 3 - 2}$.
Step 3.1.2
Rewrite the expression as $\frac{(x - 4)x + (x - 4) \cdot 2 + 3}{(x - 4)x - (x - 4) \cdot 3 - 2}$.
Step 3.2
Again, cancel the $(x - 4)$ term.
Step 3.2.1
Move the negative sign in $-\frac{2}{x - 4}$ to the numerator.
Step 3.2.2
Cancel out $(x - 4)$ to simplify $\frac{(x - 4)x + (x - 4) \cdot 2 + 3}{(x - 4)x - (x - 4) \cdot 3 + 2}$.
Step 3.2.3
Rewrite the expression as $\frac{(x - 4)x + (x - 4) \cdot 2 + 3}{(x - 4)x - (x - 4) \cdot 3 + 2}$.
Step 4
Simplify the numerator by distributing and combining like terms.
Step 5
Simplify the denominator by distributing and combining like terms.
Step 5.8
Factor the quadratic expression in the denominator.
Knowledge Notes:
Multiplying Fractions: To multiply fractions, you multiply the numerators together and the denominators together.
Distributive Property: This property states that $a(b + c) = ab + ac$. It is used to multiply a single term and two or more terms inside a set of parentheses.
Simplifying Fractions: To simplify a fraction, you divide the numerator and the denominator by their greatest common factor.
Cancelling Common Factors: When a factor appears in both the numerator and the denominator of a fraction, it can be cancelled out.
Factoring Quadratics: This involves finding two numbers that multiply to give the product of the coefficient of $x^2$ and the constant term (the "ac" in the "ac method"), and add to give the coefficient of $x$. These two numbers are then used to split the middle term and factor by grouping.
AC Method: A technique used to factor quadratics where the leading coefficient is not 1. It involves finding two numbers that multiply to the product of the leading coefficient and the constant term, and add to the middle coefficient. These numbers are used to split the middle term and factor by grouping.
