Problem 70 - Intermediate Algebra

Question

Factor the given factor from the expression. $$ x^{27} ; x^{3 / 7}-2 x^{27} $$

Step-by-Step Solution

Answer

The expression factors to \(x^{3/7}(1 - 2x^{186/7})\).

Step 1: Identify Common Factor

The given expression is \(x^{3/7} - 2x^{27}\). First, identify the common factor in both terms of the expression. The terms are \(x^{3/7}\) and \(-2x^{27}\). Here, \(x^{3/7}\) is the least power of \(x\), so the greatest common factor is \(x^{3/7}\).

Step 2: Factor Out the Common Factor

Now, factor \(x^{3/7}\) out of each term in the expression. This means we write the expression as \(x^{3/7}(1 - 2x^{27-3/7})\).

Step 3: Simplify the Exponent

Simplify the exponent of \(x\) inside the parenthesis: \(27 - \frac{3}{7} = \frac{189}{7} - \frac{3}{7} = \frac{186}{7}\). Now the expression becomes \(x^{3/7}(1 - 2x^{186/7})\).

Key Concepts

Common Factor

Factoring is a method used to break down complex algebraic expressions into simpler ones, which can be easier to work with or solve. A common factor in an expression is a term that is present in each part of the expression. In other words, it's a value that can be divided evenly from all terms involved. This is particularly useful when simplifying polynomial expressions.

To identify a common factor:

• Inspect each term in the polynomial.
• Determine a shared element among all terms, which could be a number, a variable, or a combination.
• Look for the smallest power of shared variables to ensure it's a true factor of the entire expression.

Finding the common factor helps in reducing the expression to a simpler form, which is easier to interpret or further solve. In the given exercise, the common factor is identified as \(x^{3/7}\) because it is the smallest exponent shared between the terms.

Polynomials

Polynomials are algebraic expressions made up of variables, coefficients, and exponents combined through addition, subtraction, and multiplication. They can range from simple forms like \(x + 2\) to more intricate expressions such as \(3x^4 - 2x^3 + 7x - 5\). Polynomials are categorized based on their degree, which is the highest exponent of the variable in the expression.

Understanding polynomials is crucial because they form the backbone of many algebraic operations and solve numerous practical problems. They are foundational in calculus, science, engineering, and beyond.

When working with polynomials, recognizing their structure can assist in simplifying or factoring them. Notice how in the exercise, the polynomial involved is structured with terms \(x^{3/7}\) and \(-2x^{27}\). By understanding that these are parts of a larger polynomial, you can systematically factor and manipulate them more effectively.

Exponents

Exponents express repeated multiplication of a number by itself, providing a shorthand way to denote such processes. For instance, \(x^3\) signifies \(x \times x \times x\). In algebra, managing variables with exponents is common and vital for simplifying expressions and solving equations.

Key properties of exponents include:

• Addition: \(x^a \times x^b = x^{a+b}\).
• Subtraction: \(\frac{x^a}{x^b} = x^{a-b}\), assuming \(x eq 0\).
• Multiple operations: \((x^a)^b = x^{a \times b}\).

Applying these rules helps in adjusting expressions where exponents are involved. For instance, in the exercise solution, exponents are simplified when factoring the expression, ultimately leading to \(x^{3/7}(1 - 2x^{186/7})\). Simplifying exponents correctly is critical to achieving the right form of the expression and solving polynomial equations accurately.