Problem 78 - Algebra and Trigonometry

Question

77–84 ? Factor the expression completely. Begin by factoring out the lowest power of each common factor. $$ 3 x^{-1 / 2}+4 x^{1 / 2}+x^{3 / 2} $$

Step-by-Step Solution

Answer

The completely factored expression is \( x^{-1/2}(x^2 + 4x + 3) \).

Step 1: Identify the lowest power

First, look at the exponents of the expression. You have the terms: \( 3x^{-1/2}, 4x^{1/2}, \) and \( x^{3/2} \). The exponents are \(-1/2, 1/2,\) and \(3/2\) respectively. The smallest of these is \(-1/2\).

Step 2: Factor out the lowest power

Factor out \( x^{-1/2} \) from each term:\[ 3x^{-1/2} + 4x^{1/2} + x^{3/2} = x^{-1/2}(3 + 4x + x^2) \].Here, each term in the parentheses is obtained by dividing by \( x^{-1/2} \).

Step 3: Write the factored expression

The expression is now factored as:\[ x^{-1/2}(x^2 + 4x + 3).\]

Key Concepts

Common Factors

When factoring expressions, identifying common factors is crucial. A common factor is a term or number that divides each term in an expression without leaving a remainder. To find common factors, observe each term closely:

• In the expression \(3x^{-1/2}, 4x^{1/2}, \) and \(x^{3/2}\), notice that the variable \(x\) is present in every term.
• The common factor here is the lowest exponent of \(x\): \(-1/2\).

This shared factor is what you "pull out" from the expression first. By factoring out the common factor, you simplify the original expression, making further manipulations or solving easier. In our exercise, by factoring out \(x^{-1/2}\), each term becomes simpler and results in the new expression \(x^{-1/2}(3 + 4x + x^2)\). Finding common factors is like looking for the simplest key that unlocks the whole expression, making further calculations manageable.

Exponents

Exponents indicate how many times a number or variable is multiplied by itself. They are a form of notation that makes expressions efficient and compact. When dealing with exponents, it helps to remember a few basic rules:

• When multiplying, add the exponents if the bases are the same.
• When dividing, subtract the exponents.

For example, in the exercise \(3x^{-1/2}, 4x^{1/2}, \) and \(x^{3/2}\), the expressions have different exponents: \(-1/2\), \(1/2\), and \(3/2\).
Choosing the lowest of these, \(-1/2\), helps in simplifying the expression when factoring. The act of factoring involves adjusting each term relative to this lowest exponent. For example, when factoring out \(x^{-1/2}\), each term adjusts as if \(x^{-1/2}\) is being divided out, leading to better understanding and a simplified form of the expression like \(x^{-1/2}(3 + 4x + x^2)\). Understanding exponents is key to managing and manipulating algebraic expressions effectively.

Algebraic Expressions

Algebraic expressions consist of terms that include variables, coefficients, and exponents. They form the building blocks of algebra and represent quantities or relationships in general terms. These expressions can be simplified, factored, or expanded based on rules and principles of algebra.

When working with algebraic expressions, like \(3x^{-1/2} + 4x^{1/2} + x^{3/2}\), it involves operating on each component:

• Terms: individual elements such as \(3x^{-1/2}\).
• Coefficients: the numbers before the variables, like \(3\).
• Variables and Exponents: \(x^{-1/2}\) indicates the dynamic nature and relationships affected by the variable.

Factoring algebraic expressions involves extracting common factors and simplifying the terms. In this process, you use techniques such as factoring out the smallest exponent to make the expression easier to work with, yielding something like \(x^{-1/2}(3 + 4x + x^2)\). With practice, handling these expressions becomes intuitive and a fundamental skill for more complex mathematics.