In Exercises 17–38, factor each trinomial, or state that the trinomial is prime. 2x^2+5x−3

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Identify the trinomial in the form ax^2 + bx + c. Here, a = 2, b = 5, and c = -3.

Multiply the coefficient of x^2 (a) by the constant term (c). In this case, calculate 2 * (-3) = -6.

Find two numbers that multiply to give the product from step 2 (-6) and add to give the middle coefficient (b = 5). These numbers are 6 and -1 because 6 * (-1) = -6 and 6 + (-1) = 5.

Rewrite the middle term (5x) using the two numbers found in step 3. The trinomial becomes 2x^2 + 6x - x - 3.

Group the terms into two pairs and factor each pair. For the first pair (2x^2 + 6x), factor out 2x, giving 2x(x + 3). For the second pair (-x - 3), factor out -1, giving -1(x + 3). Combine the factored terms to get (2x - 1)(x + 3).

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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 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 a and c) and add to b. Understanding this method 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 remains in its original form. This concept is important for accurately solving polynomial equations.

Quadratic Formula

The quadratic formula, x = (-b ± √(b² - 4ac)) / (2a), provides a method for finding the roots of any quadratic equation. While not directly related to factoring, it serves as a backup method to determine if a trinomial can be factored by revealing the nature of its roots. Understanding this formula is vital for solving quadratics that may not factor easily.