Question
Find the greatest common factor of the terms x Superscript n plus 1 Baseline y Superscript n and x Superscript n Baseline y Superscript n minus 2, where n is a whole number greater than 2. How can you factor the expression x Superscript n plus 1 Baseline y Superscript n Baseline plus x Superscript n Baseline y Superscript n minus 2?
Answers
GPT-4o mini
To find the greatest common factor (GCF) of the terms \( x^{n+1}y^n \) and \( x^n y^n - 2 \), let's start by identifying the individual factors of each term.
1. **Analyze the first term**:
The term \( x^{n+1}y^n \) consists of:
\[
x^{n+1}y^n = x^n \cdot xy^n
\]
Here, the factors are \( x^n \), \( xy^n \).
2. **Analyze the second term**:
The term \( x^n y^n - 2 \) can be considered as:
\[
x^n y^n - 2
\]
This term doesn't contain a \( y^n \) factor when combined with \( x^{n+1}y^n \), but we can note that both terms share the \( x^n \) factor.
To determine the GCF based on the above:
- The first term contributes \( x^{n}y^n \).
- The second term contributes a factor of \( x^n \), but not of \( y^n \) since it cannot be simplified further.
Thus, the greatest common factor (GCF) is \(\mathbf{x^n}\).
---
Next, we’ll factor the expression \( x^{n+1}y^n + x^n y^n - 2 \):
1. **Combine the terms**:
We arrange the expression:
\[
x^{n+1}y^n + x^n y^n - 2 = x^n y^n (x + 1) - 2
\]
2. **Consider factoring further**:
We have \( x^n y^n (x + 1) - 2 \).
This expression cannot factor down further neatly because \( x^n y^n (x + 1) - 2 \) does not have a common form that can be expressed using other algebraic identities easily.
Therefore, the factorization can be simplified to:
\[
\mathbf{x^n y^n (x + 1) - 2}
\]
Thus, to summarize:
- The greatest common factor of the terms \( x^{n+1}y^n \) and \( x^n y^n - 2 \) is \( \mathbf{x^n} \).
- The expression \( x^{n+1}y^n + x^n y^n - 2 \) can be factored as \( x^n y^n (x + 1) - 2 \).
1. **Analyze the first term**:
The term \( x^{n+1}y^n \) consists of:
\[
x^{n+1}y^n = x^n \cdot xy^n
\]
Here, the factors are \( x^n \), \( xy^n \).
2. **Analyze the second term**:
The term \( x^n y^n - 2 \) can be considered as:
\[
x^n y^n - 2
\]
This term doesn't contain a \( y^n \) factor when combined with \( x^{n+1}y^n \), but we can note that both terms share the \( x^n \) factor.
To determine the GCF based on the above:
- The first term contributes \( x^{n}y^n \).
- The second term contributes a factor of \( x^n \), but not of \( y^n \) since it cannot be simplified further.
Thus, the greatest common factor (GCF) is \(\mathbf{x^n}\).
---
Next, we’ll factor the expression \( x^{n+1}y^n + x^n y^n - 2 \):
1. **Combine the terms**:
We arrange the expression:
\[
x^{n+1}y^n + x^n y^n - 2 = x^n y^n (x + 1) - 2
\]
2. **Consider factoring further**:
We have \( x^n y^n (x + 1) - 2 \).
This expression cannot factor down further neatly because \( x^n y^n (x + 1) - 2 \) does not have a common form that can be expressed using other algebraic identities easily.
Therefore, the factorization can be simplified to:
\[
\mathbf{x^n y^n (x + 1) - 2}
\]
Thus, to summarize:
- The greatest common factor of the terms \( x^{n+1}y^n \) and \( x^n y^n - 2 \) is \( \mathbf{x^n} \).
- The expression \( x^{n+1}y^n + x^n y^n - 2 \) can be factored as \( x^n y^n (x + 1) - 2 \).