Problem 84 - Algebra and Trigonometry

Question

77–84 ? Factor the expression completely. Begin by factoring out the lowest power of each common factor. $$ 3 x^{-1 / 2}\left(x^{2}+1\right)^{5 / 4}-x^{3 / 2}\left(x^{2}+1\right)^{1 / 4} $$

Step-by-Step Solution

Answer

The expression factors to \( x^{-1/2}(x^2+1)^{1/4}(2x^2+3) \).

Step 1: Identify Common Factors

First, we look for common factors in each term of the expression. The terms are \(3x^{-1/2}(x^2+1)^{5/4}\) and \(x^{3/2}(x^2+1)^{1/4}\). Both terms have \((x^2+1)\) as a common factor. Let's find the smallest power of \((x^2+1)\) to factor out: the smallest power is \((x^2+1)^{1/4}\). Similarly, we have \(x^{-1/2}\) and \(x^{3/2}\) as common factors with the smaller power being \(x^{-1/2}\).

Step 2: Factor Out the Common Factors

Factoring out the smallest powers, \((x^2+1)^{1/4}\) and \(x^{-1/2}\), from the expression gives: \\[ x^{-1/2}(x^2+1)^{1/4} \left[ 3(x^2+1)^{(5/4)-(1/4)} - x^{(3/2) - (-1/2)} \right] \] \This simplifies to: \\[ x^{-1/2}(x^2+1)^{1/4} \left[ 3(x^2+1)^{4/4} - x^{4/2} \right] \] \Which further simplifies to: \\[ x^{-1/2}(x^2+1)^{1/4} [3(x^2+1) - x^2] \]

Step 3: Simplify the Expression Inside the Brackets

Now we'll simplify the expression inside the brackets: \\[ 3(x^2 + 1) - x^2 = 3x^2 + 3 - x^2 = 2x^2 + 3 \] \Therefore, the expression inside the brackets reduces to \(2x^2 + 3\).

Step 4: Write the Final Factored Expression

Finally, we write the fully factored expression: \\[ x^{-1/2}(x^2+1)^{1/4}(2x^2+3) \] \This is the expression fully factored out into its components.

Key Concepts

Common Factors

In algebra, identifying common factors is the first step in factoring expressions. A common factor is a term or set of terms that appear in every part of an expression. Finding these can simplify complex polynomial expressions.

To determine the common factors in our example expression, look at each term component separately. For the terms \(3x^{-1/2}(x^2+1)^{5/4}\) and \(x^{3/2}(x^2+1)^{1/4}\), note the appearance of \(x\) and \((x^2+1)\).

Both terms include the factor \((x^2+1)\). So, we inspect their powers: \((x^2+1)^{5/4}\) and \((x^2+1)^{1/4}\). The smaller power is \((x^2+1)^{1/4}\), which we'll factor out.

Similarly, examine the variable \(x\) within each term: \(x^{-1/2}\) and \(x^{3/2}\). The smallest exponent here is \(x^{-1/2}\). Factoring out these smallest powers can simplify the expression tremendously. This results in finding the skeleton of the expression that’s easier to resolve.

Polynomial Expressions

Polynomials are expressions consisting of variables and coefficients combined through operations of addition, subtraction, and multiplication. They form the backbone of algebraic math and can range from simple monomials like \(2x\) to complex forms such as \(x^4 + 3x^3 + 2x^2 + x + 5\).

In the problem above, \(3x^{-1/2}(x^2+1)^{5/4} - x^{3/2}(x^2+1)^{1/4}\) looks intricate because of multiple monomial expressions.

Factoring helps in breaking down a complex polynomial into simpler parts, just as we did. By factoring out \((x^2+1)^{1/4}\) and \(x^{-1/2}\), we distilled the expression to a more comprehensible form that is easy to calculate or graph.

Understanding polynomial degrees, coefficients, and the effect of operations on these expressions builds a solid foundation for solving more sophisticated algebraic problems.

Exponents Simplification

Exponents represent how many times a number (the base) is multiplied by itself. Simplifying exponents is crucial in algebra, allowing complex expressions to be transformed into simpler ones.

In the example, the exponents involve powers of \(x\) and the term \((x^2+1)\). To simplify, focus first on aligning operations of subtraction or addition of exponents:

• Subtract exponents when factoring like bases result in division, as with \((x^2+1)^{5/4} - (x^2+1)^{1/4}\).
• The same rule applies to \(x\), where \(x^{3/2} - x^{-1/2}\).

Clear handling of signs and understanding exponent rules such as \(a^m \times a^n = a^{m+n}\) or \(a^m / a^n = a^{m-n}\) are pivotal.

Through simplification, transformations like \(3(x^2+1)^{4/4} - x^{4/2}\) become manageable, truncating calculation complexities and making results accessible. This makes exponent simplification an invaluable tool in algebra.