Simplify x/(x^3+6x^2-8x)
The given problem asks you to perform a simplification on the algebraic expression \( \frac{x}{x^3 + 6x^2 - 8x} \). Simplification in this context usually means factoring common terms and potentially cancelling out terms in the numerator and denominator to reduce the expression to its simplest form. The goal is to rewrite the expression in a way that may make it easier to handle or understand, especially when solving equations or performing further algebraic operations.
$\frac{x}{x^{3} + 6 x^{2} - 8 x}$
Answer
Solution:
Step 1: Extract the common factor 'x' from the denominator.
- Step 1.1: Take 'x' out of $x^3$. $\frac{x}{x \cdot x^2 + 6x^2 - 8x}$
- Step 1.2: Remove 'x' from $6x^2$. $\frac{x}{x \cdot x^2 + x(6x) - 8x}$
- Step 1.3: Extract 'x' from $-8x$. $\frac{x}{x \cdot x^2 + x(6x) + x(-8)}$
- Step 1.4: Factor 'x' from $x \cdot x^2 + x(6x)$. $\frac{x}{x(x^2 + 6x) + x(-8)}$
- Step 1.5: Complete factoring 'x' from the entire expression. $\frac{x}{x(x^2 + 6x - 8)}$
Step 2: Simplify by canceling out the common 'x' factor.
- Step 2.1: Eliminate the common 'x'. $\frac{\cancel{x}}{\cancel{x}(x^2 + 6x - 8)}$
- Step 2.2: Present the simplified expression. $\frac{1}{x^2 + 6x - 8}$
Knowledge Notes:
The problem involves simplifying a rational expression by factoring and canceling common factors. The key knowledge points include:
Factoring: This is the process of breaking down a complex expression into simpler factors that, when multiplied together, give the original expression. In this case, we are factoring out the greatest common factor (GCF), which is 'x'.
Common Factors: When the numerator and denominator of a fraction share a common factor, they can be canceled out to simplify the expression. This is based on the property that $\frac{a \cdot c}{b \cdot c} = \frac{a}{b}$ for any non-zero 'c'.
Rational Expressions: These are fractions where the numerator and/or the denominator are polynomials. Simplifying rational expressions often involves factoring polynomials and canceling common factors.
Polynomials: A polynomial is a mathematical expression consisting of variables, coefficients, and exponents, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponentiation of variables.
Exponents: An exponent refers to the number of times a number (the base) is multiplied by itself. For example, $x^3$ means $x$ multiplied by itself three times.
LaTeX Formatting: To clearly present mathematical expressions, LaTeX is used. It is a typesetting system that is widely used for the communication and publication of scientific documents in mathematics, computer science, engineering, and physics.
By applying these concepts, the given rational expression is simplified by factoring out the common 'x' from the denominator and then canceling it with the 'x' in the numerator.
