Problem 85 - Algebra and Trigonometry

Question

Factor the expression completely. Begin by factoring out the lowest power of each common factor. $$ x^{-3 / 2}+2 x^{-1 / 2}+x^{1 / 2} $$

Step-by-Step Solution

Answer

Factored form: \(x^{-3/2}(1 + 2x + x^{7/2})\).

Step 1: Identify the Lowest Power

The expression is composed of three terms: \(x^{-3/2}\), \(2x^{-1/2}\), and \(x^{1/2}\). Start by identifying the lowest power of \(x\) in these terms, which is \(-3/2\).

Step 2: Factor Out the Lowest Power

Factor \(x^{-3/2}\) out of each term in the expression: \[x^{-3/2} (1 + 2x + x^2 x^{3/2}).\]

Step 3: Simplify the Expression Inside the Parentheses

Simplify the expression inside the parentheses: \(1 + 2x + x^2 \cdot x^{3/2}\) simplifies to \(1 + 2x + x^{7/2}\).

Step 4: Analyze for Further Factoring

Check if the expression inside the parentheses, \(1 + 2x + x^{7/2}\), can be factored further. All terms are distinct and don't contain common factors, leaving no further factorizable common terms.

Step 5: Combine and Recheck

Recombine the factored and simplified expression to confirm accuracy: \[x^{-3/2}(1 + 2x + x^{7/2}).\] This is as simplified as possible given the powers.

Key Concepts

Understanding Exponents

Exponents are a critical component of algebra and appear frequently in equations and expressions. In simple terms, an exponent denotes how many times a number, known as the base, is multiplied by itself. The expression \(x^n\), for example, means \(x\) is multiplied by itself \(n\) times.
Exponents can be positive, negative, or even fractions:

• **Positive exponents** indicate standard repeated multiplication. For instance, \(x^3 = x \times x \times x\).
• **Negative exponents** are a shorthand for division by the base. Thus, \(x^{-n} = 1/x^n\). For example, \(x^{-2} = 1/(x^2)\).
• **Fractional exponents** represent roots. So, \(x^{1/2}\) is the square root of \(x\), and \(x^{3/2}\) is \((x^{1/2})^3\) or \(\sqrt{x^3}\).

Mastering how to work with exponents, including adding and subtracting them when multiplying like bases and recognizing how negative and fractional exponents affect a base, is essential for simplifying algebraic expressions effectively.

Recognizing Polynomial Expressions

Polynomial expressions might seem complex at first glance, but they are just sums of variables raised to powers and their coefficients. A polynomial expression is generally written in the form \(a_nx^n + a_{n-1}x^{n-1} + \,\ldots\, + a_1x + a_0\), where \(a_0, a_1,\ldots, a_n\) are constants.

The expression we focused on, \(x^{-3/2} + 2x^{-1/2} + x^{1/2}\), might not initially appear like a standard polynomial due to fractional and negative exponents. However, it maintains the key characteristic of being a sum of terms, each composed of a constant (coefficient) and a variable raised to an exponent.

When working with polynomial expressions, it helps to identify the largest and smallest powers of the variable, like in the original exercise. This strategy aids in factoring and simplifying the expression down further. Recognizing the structure and components gives you the toolkit to manipulate and simplify the expression in meaningful ways.

Simplifying Algebraic Expressions

Simplifying algebraic expressions is about transforming an expression into its most concise and meaningful form. It involves combining like terms, factoring, and often breaking down more complex entities into simpler parts. This step is crucial for making equations easier to solve or insights clearer.
In the exercise, simplifying began with identifying the lowest common exponent, \(-3/2\), in the terms of the expression. By factoring this out, we simplified the expression into a product of two components: \(x^{-3/2}(1 + 2x + x^{7/2})\).
Although further simplification was not possible due to the distinct powers in the expression inside the parentheses, the factored form offers a different perspective that might simplify additional operations on the expression in broader contexts. To efficiently simplify expressions, always,

• Identify and factor out common terms or variables.
• Simplify within parentheses, if any, after factoring.
• Check if the remaining expression can be collected further or expressed more simply.

These steps ensure that you're left with the cleanest version of your expression, which is particularly useful for solving algebraic equations and facilitating other mathematical operations.