Question

which statement is true about the factorization of $30x^{2}+40xy + 51y^{2}$? the polynomial can be rewritten after factoring as $10(3x^{2}+4xy + 5y^{2})$. the polynomial can be rewritten as the product of a trinomial and $xy$. the greatest common factor of the polynomial is $51x^{2}y^{2}$. the greatest common factor of the terms is 1.

Explanation:

Step1: Analyze coefficients' GCF

Find GCF of 30, 40, 51:
Factors of 30: 1,2,3,5,6,10,15,30
Factors of 40: 1,2,4,5,8,10,20,40
Factors of 51: 1,3,17,51
GCF = 1

Step2: Analyze variable terms

Terms have $x^2$, $xy$, $y^2$; no common variable factor.

Step3: Evaluate each option

• Option1: $10(3x^2+4xy+5y^2)=30x^2+40xy+50y^2≠30x^2+40xy+51y^2$ (false)
• Option2: No common variable factor of $xy$, cannot factor with it (false)
• Option3: 51 is only coefficient of $y^2$, not a common factor (false)
• Option4: GCF of terms is 1 (true)

Answer:

The greatest common factor of the terms is 1.