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^2y^2).
• the greatest common factor of the terms is 1.

Explanation:

Find the greatest common factor of the coefficients

The coefficients of the terms in the polynomial \(30x^2 + 40xy + 51y^2\) are \(30\), \(40\), and \(51\).

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Find the greatest common factor of the variable parts

The variable parts of the terms are \(x^2\), \(xy\), and \(y^2\).

$$ \gcd(x^2, xy, y^2) = 1 $$

Determine the overall greatest common factor

Combine the coefficient GCF and the variable GCF to find the GCF of the terms.

$$ \text{GCF} = 1 \times 1 = 1 $$

Answer:

• The polynomial can be rewritten after factoring as \(30x^2 + 40xy + 51y^2\).
• The polynomial can be rewritten as the product as \(10(3x^2 + 4xy + 5y^2)\).
• The greatest common factor of the polynomial is \(51x^2y^2\).
• The greatest common factor of the terms is 1. (Correct answer)