Problem 36 - College Algebra Essentials

Question

Factor each trinomial, or state that the trinomial is prime. $$3 x^{2}+4 x y+y^{2}$$

Step-by-Step Solution

Answer

The given trinomial can be factored as \((\sqrt{3}x+\frac{2\sqrt{3}}{3}y)^{2}\).

Step 1: Identify the structure

The given trinomial is in the form of \(ax^{2} + bxy + cy^{2}\), where \(a=3\), \(b=4\) and \(c=1\). The basic idea is to express the trinomial as \((px+qy)^{2}\), which equals to \(p^{2}x^{2} + 2pqxy+ q^{2}y^{2}\). Comparing coefficients, we have \(a=p^{2}\), \(b=2pq\), and \(c=q^{2}\).

Step 2: Solve for p and q

By comparing coefficients, we can solve for p and q. From \(a=p^{2}\), we get \(p=\sqrt{3}\). Plugging into \(b=2pq\) (4=2\(\sqrt{3}q)), we solve for q to get \(q=2/\sqrt{3}\), simplifying this further gives \(q=\frac{2\sqrt{3}}{3}\). Finally, we check that \(c=q^{2}\) equals to \(1\), which holds.

Step 3: Final factorization

Finally, we replace p and q in \((px+qy)^{2}\) and get \((\sqrt{3}x+\frac{2\sqrt{3}}{3}y)^{2}\). This verifies that the trinomial \(3x^{2}+4xy+y^{2}\) can be expressed as \((\sqrt{3}x+\frac{2\sqrt{3}}{3}y)^{2}\).

Key Concepts

Quadratic Expressions

Quadratic expressions are the cornerstone of many algebraic concepts and are widely used in mathematical equations. These expressions typically take the form of \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(x\) represents a variable. In our example, the expression is slightly more complex and is represented as \(ax^2 + bxy + cy^2\). Here, both \(x\) and \(y\) are variables, allowing for a broader range of applications. Quadratic expressions often describe parabolas in geometry and can model physical phenomena such as projectile motion.

Learning to factor these expressions into the product of two binomials is a key skill in algebra. Factoring involves rewriting the quadratic expression in a simplified form that still equates to the original expression when expanded. Besides simplifying expressions, factoring can also be used to solve quadratic equations and find roots. This form of expression can greatly simplify complex algebraic operations and is vital in higher mathematics and applications.

Coefficient Comparison

Coefficient comparison is a technique used when dealing with algebraic expressions, particularly in systems of equations or when factoring polynomials.

In the original exercise, the trinomial \(3x^2 + 4xy + y^2\) is compared against a perfect square binomial \((px + qy)^2\). This is expanded to \(p^2x^2 + 2pqxy + q^2y^2\). By setting the coefficients of corresponding terms equal—namely \(a = p^2\), \(b = 2pq\), and \(c = q^2\)—we establish equations that we can solve for \(p\) and \(q\).

Here's the breakdown of this approach:

• Identify corresponding coefficients in the target trinomial and the squared binomial.
• Set up equations for each comparable coefficient.
• Solving these equations yields potential values for variables \(p\) and \(q\).

Using this method allows you to systematically work through complex expressions and find insightful solutions, often revealing the underlying structure of the equation.

Algebraic Equations

Algebraic equations involve setting expressions equal to one another and solving for the unknowns by manipulating the equations. These forms are central in algebra and have applications ranging from basic problem-solving to complex mathematical modeling.

When factoring trinomials or solving quadratic equations, we often manipulate algebraic equations. In the provided step-by-step solution, we solve an algebraic equation derived from comparing coefficients to determine the values of \(p\) and \(q\):

• From \(a=p^2\), solve for \(p\).
• Substitute \(p\) into the equation \(b=2pq\) to find \(q\).

These steps illustrate how manipulation of algebraic equations helps us progress toward solving real-world problems. By expressing the trinomial as \((px + qy)^2\), we simplify the equation into a more approachable form, enabling further analysis and solution.

Understanding and applying the concepts of algebraic equations is essential in algebra, providing a foundation for more advanced studies.