Problem 49 - College Algebra with Corequisite Support

Question

For the following exercises, factor the polynomials. \(5 z(2 z-9)^{-\frac{3}{2}}+11(2 z-9)^{-\frac{1}{2}}\)

Step-by-Step Solution

Answer

The polynomial factors to \((2z-9)^{-\frac{1}{2}} \left( \frac{5z}{2z-9} + 11 \right)\).

Step 1: Identify Common Factor

Look at the given polynomial: \[5 z(2 z-9)^{-\frac{3}{2}} + 11(2 z-9)^{-\frac{1}{2}}\]Notice that both terms have the common factor \((2z-9)^{-rac{1}{2}}\).

Step 2: Factor Out Common Term

Factor \((2z-9)^{-\frac{1}{2}}\) from both terms:\[(2z-9)^{-\frac{1}{2}} \left( 5z(2z-9)^{-1} + 11 \right)\]

Step 3: Simplify the Expression

Further simplify the expression inside the parenthesis:First, rewrite \((2z-9)^{-1}\) as \(\frac{1}{2z-9}\): \[(2z-9)^{-\frac{1}{2}} \left( \frac{5z}{2z-9} + 11 \right)\]Since there are no like terms inside the parenthesis that can be combined directly, the expression in the parenthesis remains as it is.

Key Concepts

Common Factor

When working with polynomials, finding the common factor is like searching for a shared element in a puzzle. It's the term that appears in each part of the expression, allowing you to simplify the polynomial more easily. In the given polynomial expression, we identify the common factor by examining each term. For our exercise, both terms in the expression contain the common factor \((2z-9)^{-1/2}\).

• The term \(5z(2z-9)^{-3/2}\) has \((2z-9)^{-1/2}\) as part of its factorization.
• Similarly, the term \(11(2z-9)^{-1/2}\) also includes \((2z-9)^{-1/2}\).

By factoring out \((2z-9)^{-1/2}\), we simplify the polynomial, making it easier to handle. Identifying and extracting the common factor is an essential skill in algebra, as it can help reduce complex expressions into simpler, more manageable forms.

Polynomial Expression

A polynomial expression is a mathematical phrase that can have terms involving variables and coefficients, along with operations such as addition and multiplication. In the context of our exercise, we are dealing with a polynomial that includes terms with powers of a binomial, which makes it a bit tricky.
When analyzing a polynomial, consider these elements to better understand it:

• Terms: Each part of the expression separated by a plus or minus sign is a term. In our scenario, \(5z(2z-9)^{-3/2}\) and \(11(2z-9)^{-1/2}\) are the two main terms.
• Coefficients: These are the numerical parts that multiply the variable or expression in each term. So, coefficients are \(5z\) and \(11\) in the terms above.
• Variable: This is the symbol that represents an unknown number – in this case, \(z\).

Understanding these components helps in organizing and simplifying polynomial expressions more effectively.

Simplifying Expressions

Simplifying expressions is all about making a mathematical expression easier to handle, while preserving its original value. Consider it akin to cleaning up clutter to create a neat and organized space. In algebra, this often involves:

• Factoring: Recognizing and pulling out common factors from the terms.
• Combining like terms: Adding or subtracting coefficients of terms with the same variables and exponents.

In our problem, once we factor out \((2z-9)^{-1/2}\), we are left with a simpler expression \((2z-9)^{-1/2} \left( \frac{5z}{2z-9} + 11 \right)\). We rewrote \((2z-9)^{-1}\) as \(\frac{1}{2z-9}\) to make the expression inside the parenthesis more understandable, though no additional simplification can be done.
Learning how to simplify expressions not only makes solving problems easier but also enhances your understanding of the mathematical relationships within an expression. It’s a fundamental skill that supports more advanced algebraic concepts.