Question
Factor the given polynomial by finding the greatest common monomial factor (or the negative of the greatest common monomial \[ 6 x^{3} y+2 x y^{3}-26 x y \]
Ask by Chang Christensen. in the United States
Feb 20,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The greatest common monomial factor is \(2xy\), and the factored form is \(2xy(3x^{2}+y^{2}-13)\).
Solution
Factor the expression \( 6x^3y+2xy^3-26xy \).
Factor the expression by following steps:
- step0: Factor:
\(6x^{3}y+2xy^{3}-26xy\)
- step1: Rewrite the expression:
\(2xy\times 3x^{2}+2xy\times y^{2}-2xy\times 13\)
- step2: Factor the expression:
\(2xy\left(3x^{2}+y^{2}-13\right)\)
The greatest common monomial factor of the given polynomial \(6x^{3}y+2xy^{3}-26xy\) is \(2xy\), and the factored form is \(2xy(3x^{2}+y^{2}-13)\).
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Simplify this solution Beyond the Answer
To factor the polynomial \( 6 x^{3} y + 2 x y^{3} - 26 x y \), start by identifying the greatest common monomial factor (GCF) of the terms. The coefficients 6, 2, and -26 have a GCF of 2. The variable factors include \( x \) and \( y \); the lowest powers of \( x \) and \( y \) across all terms are \( x^1 \) and \( y^1 \), respectively. Therefore, the GCF of the entire expression is \( 2xy \). Now, factor out \( 2xy \) from each term: \[ 6 x^{3} y + 2 x y^{3} - 26 x y = 2xy(3x^{2} + y^{2} - 13). \] Now the polynomial has been factored into \( 2xy(3x^{2}+y^{2}-13) \).
