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Chapter 5: Problem 84

Factor each polynomial completely. $$5 x^{3} y^{2}-y^{5}$$

Short Answer

Expert verified
\(y^2(5x^3 - y^3)\)

Step by step solution

01

- Identify Common Factors

First, identify the greatest common factor (GCF) in both terms. The given polynomial is \(5x^3y^2 - y^5\). The GCF of these terms is \(y^2\) because it is the highest power of \(y\) that divides both terms.
02

- Factor out the GCF

Factor out the greatest common factor \(y^2\) from the polynomial. This gives us: \(y^2(5x^3 - y^3)\).
03

- Factor the Remaining Polynomial

Now, look at the remaining polynomial inside the parentheses, \(5x^3 - y^3\). Since \(5x^3 - y^3\) does not factor further in terms of integers and simple algebraic operations, we leave it as it is.
04

- Combine the Factors

Combine the factors to get the final factored form: \(y^2(5x^3 - y^3)\). The polynomial is now completely factored.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Greatest Common Factor (GCF)
When tackling polynomial problems, identifying the greatest common factor (GCF) is the first crucial step. The GCF is the highest number or variable power that divides all terms in the polynomial without leaving a remainder.

In our given polynomial, \(5x^3y^2 - y^5\), we start by looking for common factors in each term.
  • The common variable here is \(y\), which is present in both terms.
  • The highest power of \(y\) that can be factored out from both terms is \(y^2\).
Identifying the GCF simplifies future steps and helps make the polynomial easier to factor completely.
Factoring Out
After identifying the greatest common factor (GCF), the next step is to 'factor out' this term from the polynomial. Factoring out means dividing each term in the polynomial by the GCF and placing the GCF outside of the remaining terms in parentheses.

This effectively breaks down the polynomial into simpler components. For example:

In \(5x^3y^2 - y^5\), we factor out the GCF \(y^2\):
\ y^2(5x^3 - y^3) \
Now, we've placed \ y^2 \ outside the parentheses, and the remaining terms inside the parentheses no longer share the same factors.
Factoring out not only simplifies the polynomial, but also prepares it for further factorization if possible.
Polynomial Simplification
Polynomial simplification involves breaking down complex expressions into their simplest, most manageable forms. This can help solve, analyze, or graph polynomials more easily.

Continuing with our example after factoring out the GCF:
\ y^2(5x^3 - y^3) \
The next step is to check whether the polynomial inside the parentheses can be factored further. Here, \(5x^3 - y^3\) doesn鈥檛 factor further in terms of integers or simple algebraic methods.

For other polynomials, you might need to:
  • Use methods like grouping or quadratic factoring.
  • Check for patterns like difference of squares.
Simplifying polynomials helps in understanding their structure, making them easier to work with in algebraic manipulations and applications.

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