In Exercises 17–38, factor each trinomial, or state that the trinomial is prime. 3x²+4xy+y²

Verified step by step guidance

1. Identify the trinomial: \(3x^2 + 4xy + y^2\). This is a quadratic trinomial in terms of \(x\) and \(y\). The goal is to factor it into two binomials, if possible.
2. Check if the trinomial is in standard form: \(ax^2 + bxy + cy^2\). Here, \(a = 3\), \(b = 4\), and \(c = 1\).
3. Multiply \(a\) and \(c\): \(3 \times 1 = 3\). Now, find two numbers that multiply to \(3\) and add to \(b = 4\). These numbers are \(3\) and \(1\).
4. Rewrite the middle term \(4xy\) using the two numbers found: \(3x^2 + 3xy + xy + y^2\). This step splits the middle term to facilitate factoring by grouping.
5. Group the terms in pairs and factor each group: \((3x^2 + 3xy) + (xy + y^2)\). Factor out the greatest common factor (GCF) from each group: \(3x(x + y) + y(x + y)\). Finally, factor out the common binomial \((x + y)\): \((3x + y)(x + y)\).

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 coefficient of x^2 and the constant term) and add to b (the coefficient of x). Understanding this concept is essential for simplifying expressions and solving equations.

Prime Trinomials

A prime trinomial is a quadratic expression that cannot be factored into the product of two binomials with rational coefficients. Recognizing when a trinomial is prime is crucial, as it indicates that the expression cannot be simplified further. This concept helps in determining the nature of the roots of the quadratic equation associated with the trinomial.

The Discriminant

The discriminant, given by the formula b^2 - 4ac for a quadratic equation ax^2 + bx + c, provides insight into 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 factorability of trinomials.