In Exercises 17–38, factor each trinomial, or state that the trinomial is prime. 6x^2−7xy−5y^2

Verified step by step guidance

1. Step 1: Identify the trinomial as a quadratic trinomial of the form ax^2 + bx + c. In this case, a = 6, b = -7, and c = -5.
2. Step 2: Look for two numbers that multiply to give ac (which is 6*-5 = -30) and add up to give b (which is -7).
3. Step 3: The two numbers that satisfy these conditions are -10 and 3. So, we can rewrite the middle term of the trinomial as -10xy + 3xy.
4. Step 4: Now, the trinomial becomes 6x^2 - 10xy + 3xy - 5y^2. We can factor by grouping, which involves factoring out the greatest common factor from each of the two groups.
5. Step 5: The first group has a common factor of 2x, and the second group has a common factor of y. So, the factored form of the trinomial is 2x(3x - 5y) + y(3x - 5y). Since the two terms now have a common factor of (3x - 5y), we can factor that out to get the final factored form: (3x - 5y)(2x + y).

Key Concepts

Factoring Trinomials

Factoring trinomials involves rewriting a quadratic expression in the form ax^2 + bx + c as a product of two binomials. This process requires identifying two numbers that multiply to ac (the product of the leading coefficient and the constant term) and add to b (the middle coefficient). Understanding this concept is essential for simplifying expressions and solving equations.

Prime Trinomials

A trinomial is considered prime if it cannot be factored into the product of two binomials with rational coefficients. Recognizing prime trinomials is crucial because it helps determine whether a quadratic expression can be simplified further or if it must be left in its original form. This concept is important for accurately solving polynomial equations.

The Discriminant

The discriminant, given by the formula b^2 - 4ac for a quadratic equation ax^2 + bx + c, helps determine the nature of the roots of the equation. If the discriminant is positive, there are two distinct real roots; if it is zero, there is one real root; and if negative, the roots are complex. Understanding the discriminant aids in analyzing the behavior of quadratic functions and their factorizability.