Question
-5x^2y+20xy-15xy^3
Answers
Answered by
GPT-4o mini
To simplify the expression \(-5x^2y + 20xy - 15xy^3\), we can factor out the greatest common factor (GCF) from all the terms.
1. Identify the GCF for the coefficients: The coefficients are -5, 20, and -15. The GCF of these numbers is 5.
2. Identify the common variable factors: Each term contains at least one \(y\). The lowest power of \(y\) among the terms is \(y\).
So, we can factor out \(-5y\) (since we want to keep the leading term negative) from the entire expression:
\[
-5y(x^2 - 4x + 3y^2)
\]
Next, we can factor the quadratic expression \(x^2 - 4x + 3y^2\):
To factor \(x^2 - 4x + 3y^2\), we need to find two numbers that multiply to \(3y^2\) (the constant term) and add up to \(-4\) (the coefficient of \(x\)). The numbers \(-1\) and \(-3\) work:
\[
x^2 - 4x + 3y^2 = (x - 1)(x - 3y)
\]
Putting it all together, we end up with:
\[
-5y(x - 1)(x - 3y)
\]
Therefore, the factored form of the original expression \(-5x^2y + 20xy - 15xy^3\) is:
\[
-5y(x - 1)(x - 3y)
\]
1. Identify the GCF for the coefficients: The coefficients are -5, 20, and -15. The GCF of these numbers is 5.
2. Identify the common variable factors: Each term contains at least one \(y\). The lowest power of \(y\) among the terms is \(y\).
So, we can factor out \(-5y\) (since we want to keep the leading term negative) from the entire expression:
\[
-5y(x^2 - 4x + 3y^2)
\]
Next, we can factor the quadratic expression \(x^2 - 4x + 3y^2\):
To factor \(x^2 - 4x + 3y^2\), we need to find two numbers that multiply to \(3y^2\) (the constant term) and add up to \(-4\) (the coefficient of \(x\)). The numbers \(-1\) and \(-3\) work:
\[
x^2 - 4x + 3y^2 = (x - 1)(x - 3y)
\]
Putting it all together, we end up with:
\[
-5y(x - 1)(x - 3y)
\]
Therefore, the factored form of the original expression \(-5x^2y + 20xy - 15xy^3\) is:
\[
-5y(x - 1)(x - 3y)
\]