Problem 47 - College Algebra with Corequisite Support

Question

For the following exercises, factor the polynomials. \(14 x(x+2)^{-\frac{2}{5}}+5(x+2)^{\frac{3}{5}}\)

Step-by-Step Solution

Answer

Factor the common term and simplify to get \( (x+2)^{\frac{3}{5}} \cdot \frac{19x + 10}{x+2} \).

Step 1: Identify the Common Factor

To factor the polynomial, first look for any common factors in each term. In this case, we can observe that both terms share a common factor of \((x + 2)^{\frac{3}{5}}\).

Step 2: Factor Out the Common Term

Factor \((x + 2)^{\frac{3}{5}}\) from each term. Rewrite the polynomial as:\[(x + 2)^{\frac{3}{5}} \left[ 14x(x + 2)^{-1} + 5 \right]\]

Step 3: Simplify Inside the Brackets

Simplify the expression inside the brackets by resolving the exponent. Since \((x + 2)^{-1} = \frac{1}{x + 2}\), the expression becomes:\[(x + 2)^{\frac{3}{5}} \left[ \frac{14x}{x + 2} + 5 \right]\]

Step 4: Simplify the Bracket Expression

To combine inside the brackets, find a common denominator, which is \(x + 2\). Therefore:\[(x + 2)^{\frac{3}{5}} \left[ \frac{14x + 5(x + 2)}{x + 2} \right]\]Expand and simplify the numerator:\[ 14x + 5x + 10 = 19x + 10 \]

Step 5: Final Simplified Form

After simplifying, the factored form of the polynomial is:\[(x + 2)^{\frac{3}{5}} \cdot \frac{19x + 10}{x + 2}\]Given the division, we can conclude that this is the simplest factored form for the polynomial.

Key Concepts

Common Factor

Identifying a common factor is often the first and vital step in polynomial factoring. In the example provided, the given polynomial is \(14x(x+2)^{-\frac{2}{5}} + 5(x+2)^{\frac{3}{5}}\). Spotting a common factor involves looking for similar parts in each term of the polynomial. Here, both terms share the factor \((x+2)^{\frac{3}{5}}\).
By factoring this out, we simplify the rest of the expression. This process is akin to finding the greatest common divisor in arithmetic. By pulling out the common factors, we reduce the complexity of the expression, making further simplification possible. Always remember, finding a common factor can considerably ease the work ahead in polynomial manipulation.

Exponent Simplification

Exponent simplification is crucial in managing expressions with powers. In the original problem, terms with exponents were simplified to help factor the polynomial.
Understanding exponent rules is key:

• Negative exponents indicate reciprocal: \((a^{-n} = \frac{1}{a^n})\).
• Exponents with fractions indicate roots or rational expressions.

In the problem, simplifying \((x + 2)^{-1}\) to \(\frac{1}{x + 2}\) helped achieve a simplified expression inside the brackets. This allowed the terms to be more easily combined and factored. By mastering these exponent rules, resolving complex expressions becomes a structured task.

Polynomial Simplification

Simplifying polynomials involves several techniques, one of them being combining like terms. In the exercise, further simplification occurred after factoring out the common term. Specifically, inside the brackets, the expression was simplified by performing operations over common denominators. To do this:

• Identify similar terms.
• Find a common denominator, e.g., \(x+2\) in this case.
• Combine terms appropriately, such as expanding and summing: \(14x + 5x + 10 = 19x + 10\).

This step is essential to attain the simplest form of the polynomial, enabling an easier view of its factorability and underlying structure.

Algebraic Expressions

Algebraic expressions form the backbone of mathematics and often require manipulation through techniques like factoring. Understanding and handling these expressions is crucial in a variety of math problems.
Each term in an algebraic expression involves coefficients, variables, and possibly exponents. The key to simplifying and solving problems involving these expressions is realizing how to rearrange and factor them efficiently.

• Identify terms and their components.
• Utilize proper math rules (factorization, distributive property).
• Simplify step by step to reduce the complexity.

Grasping the nature of algebraic expressions, as in this exercise, is a cornerstone for mastering more advanced algebra topics.