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Chapter 9: Problem 20

Reduce each of the following fractions as completely as possible. $$\frac{4 x+8}{7 x+14}$$

Short Answer

Expert verified
\(\frac{4}{7}\)

Step by step solution

01

Factor out the common term in the numerator

Look at the numerator, which is \(4x + 8\). Both terms have a common factor of 4. Factor out the 4 to get \(4(x + 2)\).
02

Factor out the common term in the denominator

Now look at the denominator, which is \(7x + 14\). Both terms have a common factor of 7. Factor out the 7 to get \(7(x + 2)\).
03

Simplify the fraction

Substitute the factored expressions back into the fraction: \(\frac{4(x + 2)}{7(x + 2)}\). Notice that \(x + 2\) is present in both the numerator and the denominator. Cancel \(x + 2\) from both to get \(\frac{4}{7}\).

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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 crucial skill in algebra. It involves breaking down a polynomial into simpler 'factor' polynomials that, when multiplied together, yield the original polynomial. For example, in the expression \(4x + 8\), both terms share a common factor of 4. Factoring out the 4 gives us \(4(x + 2)\). This means we rewrite the polynomial as a product of 4 and \(x + 2\). Understanding how to factor polynomials simplifies managing complex expressions and solving equations.
Algebraic Fractions
Algebraic fractions are fractions where the numerator and/or the denominator contain algebraic expressions. Consider the fraction \(\frac{4x+8}{7x+14}\). Unlike simple fractions, you may need to perform additional steps like factoring polynomials to simplify them. By managing algebraic fractions correctly, you can simplify complex algebraic expressions and solve equations more effectively. Keep in mind that any factor common to both the numerator and the denominator can be factored out to simplify the structure of the fraction.
Canceling Common Factors
Canceling common factors is a simplification technique applied to fractions. When you have common factors in both the numerator and the denominator, you can 'cancel' them, or remove them from the fraction entirely. For instance, in the previously factored form \(\frac{4(x+2)}{7(x+2)}\), the term \(x+2\) appears in both the numerator and the denominator. By canceling out \(x+2\), the fraction simplifies to \(\frac{4}{7}\). This process is essential for reducing fractions to their simplest form. Always ensure you've factored completely before canceling any terms.
Numerator and Denominator
Understanding the roles of the numerator and the denominator is fundamental in working with fractions. The numerator is the top part of the fraction--it tells us how many parts we have out of the whole. The denominator is the bottom part of the fraction--it tells us the total number of equal parts. In our example, \(4x+8\) is the numerator and \(7x+14\) is the denominator. By factoring out common terms (numerator: 4, denominator: 7) and simplifying, we can reduce the fraction to \(\frac{4}{7}\), making it easier to understand and work with.

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Most popular questions from this chapter

Perform the indicated operations and simplify as completely as possible. $$\left(\frac{x}{3}-\frac{x}{4}\right) \cdot \frac{12}{x^{2}-3 x}$$

If the exercise is an equation, solve it and check. Otherwise, perform the indicated operations and simplify. $$\frac{5}{y^{2}-3 y-10}+\frac{3}{y-5}=\frac{-1}{4 y+8}$$

Solve for \(x: \frac{x}{3}=\frac{x+6}{2}\).

In Exercises \(1-36,\) solve each of the equations or inequalities explicitly for the indicated variable. $$4(x-y)-\frac{x-y}{2}=3-(y-x) \text { for } x$$

In Exercises \(1-36,\) solve each of the equations or inequalities explicitly for the indicated variable. $$4(x-y)-\frac{x-y}{2}=3-(y-x) \text { for } y$$
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