Question
Factor the expression completely x^3y^2 + x^2y + x
Answers
Answered by
GPT-4o mini
To factor the expression \( x^3y^2 + x^2y + x \), we can first look for common factors in the terms.
1. **Identify the common factor**: Each term in the expression has \( x \) as a common factor. We can factor \( x \) out:
\[
x^3y^2 + x^2y + x = x(x^2y^2 + xy + 1)
\]
2. **Factor the remaining expression**: Now we need to see if we can factor the expression inside the parentheses, \( x^2y^2 + xy + 1 \). To check if this quadratic in terms of \( xy \) factors nicely, we can apply the method of looking for two numbers that multiply to \( 1 \) (the constant term) and add to \( 1 \) (the coefficient of \( xy \)).
Unfortunately, there are no such integer pairs for this quadratic, so it cannot be factored further using integer coefficients.
Thus, the complete factored form of the expression is:
\[
\boxed{x(x^2y^2 + xy + 1)}
\]
This is the simplest form we can achieve.
1. **Identify the common factor**: Each term in the expression has \( x \) as a common factor. We can factor \( x \) out:
\[
x^3y^2 + x^2y + x = x(x^2y^2 + xy + 1)
\]
2. **Factor the remaining expression**: Now we need to see if we can factor the expression inside the parentheses, \( x^2y^2 + xy + 1 \). To check if this quadratic in terms of \( xy \) factors nicely, we can apply the method of looking for two numbers that multiply to \( 1 \) (the constant term) and add to \( 1 \) (the coefficient of \( xy \)).
Unfortunately, there are no such integer pairs for this quadratic, so it cannot be factored further using integer coefficients.
Thus, the complete factored form of the expression is:
\[
\boxed{x(x^2y^2 + xy + 1)}
\]
This is the simplest form we can achieve.
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