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Chapter 7: Problem 13

\(\frac{6 x}{x^{2}-7 x}-\frac{2 x-x^{2}}{x^{2}-7 x}\)

Short Answer

Expert verified
The simplified form of the given expression is \(\frac{x + 4}{x - 7}\)

Step by step solution

01

Identify Like Terms

The initial expression is \(\frac{6 x}{x^{2}-7 x}-\frac{2 x-x^{2}}{x^{2}-7 x}\). Since the denominators of the fractions are the same, the problem can be simplified by combining the numerators.
02

Combine Numerators

When combining the numerators, \(6x - (2x - x^2)\) becomes \(6x - 2x + x^2\), which simplifies to \(x^2 + 4x\). The expression now becomes: \(\frac{x^2 + 4x}{x^2 - 7x}\).
03

Simplify Expression

Examine the numerator and the denominator for common factors. Both the numerator and denominator share \(x\) as a common factor. Factoring out \(x\) from both the numerator and the denominator, we get: \(\frac{x(x + 4)}{x(x - 7)}\).
04

Cancel Redundant Terms

Here, \(x\) occurs in both the numerator and the denominator, so it can be cancelled out. So the simplified fraction is: \(\frac{x + 4}{x - 7}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Factoring Polynomials
Factoring polynomials is a fundamental concept that is frequently used in algebra. The process involves breaking down a polynomial into simpler "factors" that, when multiplied together, yield the original polynomial. Let's consider the example from the exercise:
  • The expression \(x^2 + 4x \) in the numerator can be factored by finding the greatest common factor (GCF), which is \(x\).
  • This allows us to rewrite the polynomial as \(: x(x + 4) \).
Factoring is essential because it simplifies more complex algebraic fractions and helps in solving equations. It is crucial to practice identifying common factors to become efficient at factoring polynomials.
Simplifying Expressions
Simplifying expressions involves reducing them to their simplest form so they are easier to work with. This often requires factoring as well as cancelling redundant terms. In our example, we start with the expression \[\frac{x^2 + 4x}{x^2 - 7x}\] and factor both the numerator and the denominator:
  • The numerator factors to \(x(x + 4) \).
  • The denominator, \(x^2 - 7x \), factors to \(x(x - 7) \).
After factoring, we can cancel the common factor \(x\) present in both the numerator and the denominator. This gives us the simplified expression \(\frac{x + 4}{x - 7}\).
Simplifying expressions helps make complex problems manageably smaller and often reveals hidden solutions in equations.
Like Terms
In algebra, combining like terms is an essential step in simplifying expressions. Like terms are terms that contain the same variables raised to the same power. For example, \(6x\) and \(-2x\) can be combined because they both involve the term \(x\).
When simplifying the initial equation \(\frac{6x}{x^{2}-7x}-\frac{2x-x^{2}}{x^{2}-7x}\), the like terms in the numerators (terms involving \(x\)) can be combined to create a simpler, single equation.
  • We have: \(6x - (2x - x^2)\), which simplifies to \(x^2 + 4x \).
Understanding how to identify and combine like terms is crucial because it reduces the complexity of mathematical expressions, making them easier to understand and solve.

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Most popular questions from this chapter

Vocabulary What is an extraneous solution? How do you identify an extraneous solution?

. \(\frac{-2 x}{x-1}+2=\frac{10}{x-1}\)

The equation \(1-\frac{8}{x-5}=\frac{3}{x}\) has an extraneous solution of \(5 .\)

\(3 x^{3}+6 x^{2}+2 x+4\)

\(\frac{2 x}{3}\) and \(\frac{3 x}{5}\)
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