Factor each trinomial, or state that the trinomial is prime. $6 x^{2} - 7 x y - 5 y^{2}$

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Identify the trinomial to factor: $6x^{2} - 7xy - 5y^{2}$.

Multiply the coefficient of $x^{2}$ (which is 6) by the coefficient of $y^{2}$ (which is -5), giving $6 \times (-5) = -30$.

Find two numbers that multiply to -30 and add to the middle coefficient, which is -7. These numbers are -10 and 3 because $-10 \times 3 = -30$ and $-10 + 3 = -7$.

Rewrite the middle term $-7xy$ as $-10xy + 3xy$ to split the trinomial: $6x^{2} - 10xy + 3xy - 5y^{2}$.

Group the terms in pairs and factor each group: from $6x^{2} - 10xy$, factor out \$2x$ to get \(2x(3x - 5y)$; from $3xy - 5y^{2}$, factor out \)y$ to get $y(3x - 5y)$. Then factor out the common binomial $(3x - 5y)$ to get $(3x - 5y)(2x + y)$.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Factoring Trinomials

Factoring trinomials involves expressing a quadratic expression as a product of two binomials. For trinomials in two variables, such as 6x²−7xy−5y², the goal is to find two binomials whose product equals the original expression. This process often requires identifying pairs of terms that multiply to the constant term and add to the middle term.

Prime Trinomials

A trinomial is prime if it cannot be factored into the product of two binomials with integer coefficients. Determining whether a trinomial is prime involves attempting to factor it and, if no suitable factors exist, concluding that it is prime. Recognizing prime trinomials helps avoid unnecessary factoring attempts.

Factoring by Grouping

Factoring by grouping is a method used when the trinomial's leading coefficient is not 1. It involves splitting the middle term into two terms whose coefficients multiply to the product of the leading coefficient and the constant term. Then, grouping terms and factoring out common factors from each group can lead to the factorization of the entire trinomial.

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