Problem 48 - College Algebra with Corequisite Support

Question

For the following exercises, factor the polynomials. \(9 y(3 y-13)^{\frac{1}{5}}-2(3 y-13)^{\frac{6}{5}}\)

Step-by-Step Solution

Answer

The factored expression is \((3y-13)^{\frac{1}{5}}(3y + 26)\).

Step 1: Identify Common Factors

First, observe the expression \(9 y(3 y-13)^{\frac{1}{5}} - 2(3 y-13)^{\frac{6}{5}}\). Notice that both terms involve the expression \((3y-13)\). Specifically, the powers are \((3y-13)^{\frac{1}{5}}\) and \((3y-13)^{\frac{6}{5}}\). We can factor out the smallest power, \((3y-13)^{\frac{1}{5}}\), as the common factor.

Step 2: Factor out the Common Term

Factor \((3y-13)^{\frac{1}{5}}\) from the entire expression: \[ (3y-13)^{\frac{1}{5}} \left( 9y - 2\frac{(3y-13)^{\frac{6}{5}}}{(3y-13)^{\frac{1}{5}}} \right) \]Simplifying the inner expression, we get: \[ (3y-13)^{\frac{1}{5}}(9y - 2 (3y-13)) \]

Step 3: Simplify the Remaining Expression

Now simplify the expression inside the parentheses: \[ 9y - 2(3y-13) = 9y - 6y + 26 \]Combine like terms:\[ 3y + 26 \]

Step 4: Write the Final Factored Form

The expression can be rewritten by multiplying the common factor with the simplified expression:\[ (3y-13)^{\frac{1}{5}}(3y + 26) \] This is the fully factored form of the given polynomial expression.

Key Concepts

Common Factors

When working with polynomials, identifying common factors is the first, crucial step in factorizing the expression. A common factor is a number or expression that divides all terms of a polynomial evenly. By factoring it out, we simplify the expression and make it more manageable to work with.
In our example, both terms in the polynomial \(9y(3y-13)^{\frac{1}{5}} - 2(3y-13)^{\frac{6}{5}}\) share the factor \((3y-13)\). The exponents associated with this common term, \(\frac{1}{5}\) and \(\frac{6}{5}\), indicate the powers to which the common factor is raised. To factor properly, always choose the smallest exponent of the repeated factor to be taken out of the expression.
**Why Factoring Common Factors Matters:** Factorizing simplifies further calculations and reveals additional algebraic properties, like roots or symmetry.

• Reduces complexity.
• Provides a clearer algebraic form.
• Essential for solving polynomial equations.

Exponents

Exponents are a fundamental part of polynomials, representing repeated multiplication of a base. In polynomial expressions, exponents dictate the degree and nature of the polynomial.
For example, consider the terms \((3y - 13)^{\frac{1}{5}}\) and \((3y - 13)^{\frac{6}{5}}\). In these terms, \((3y-13)\) acts as the base while \(\frac{1}{5}\) and \(\frac{6}{5}\) are the exponents. These fractional exponents can seem daunting, but they follow the same basic rules as integer exponents.
**Working with Exponents in Factorization:**

• Use properties of exponents: \((a^m \cdot a^n = a^{m+n})\) to simplify terms.
• Choose the lower exponent to factor out common bases from terms.
• This allows simplification and easy handling of the expression.

By mastering these rules, you can simplify complex polynomial expressions with confidence.

Simplifying Expressions

Simplification is the key to easier manipulation and understanding of algebraic expressions. After identifying and factoring out a common factor, simplifying the polynomial inside the parentheses is the next step. This process involves combining like terms and solving any remaining operations.
In our example, after factoring out \((3y-13)^{\frac{1}{5}}\), the expression becomes \[9y - 2\frac{(3y-13)^{\frac{6}{5}}}{(3y-13)^{\frac{1}{5}}}\].
Simplify this to \[9y - 2(3y-13)\]. With distribution, this becomes \[9y - 6y + 26\]. Combine the like terms, \[3y + 26\], to achieve the simplest form.
**Why Simplifying is Essential:**

• Simplifying polynomials reveals their true behavior and qualities.
• It provides a clearer path to solving equations or further factorization.
• It's an essential step in many algebraic processes.

Ensuring that each expression you work with is in its simplest form is crucial for correct and efficient algebraic manipulation.

Polynomial Expressions

Polynomial expressions are algebraic equations involving a sum of powers of one or more variables multiplied by coefficients. These equations are foundational in algebra, and understanding their structure and behavior is necessary for factorization.
Consider the polynomial \(9y(3y-13)^{\frac{1}{5}} - 2(3y-13)^{\frac{6}{5}}\) we previously discussed. This expression showcases all typical components of polynomials: terms, coefficients like 9 and -2, variables like \(y\), and exponents.
**Characteristics of Polynomial Expressions:**

• They may include one or more terms.
• Each term can have constants, variables, and exponents.
• Typically written in standard form, terms are ordered from high to low exponents.

Understanding these components helps when factoring and simplifying, as each aspect of the polynomial holds significance in the context of algebra.
Factoring and handling polynomials effectively depends on recognizing these elements and applying algebraic principles to manipulate them.