Problem 13 - Algebra A Combined Function

Question

Factor each trinomial completely. If a polynomial can't be factored, write "prime." $$ x^{2}+8 x y+15 y^{2} $$

Step-by-Step Solution

Answer

The trinomial factors are \((x + 3y)(x + 5y)\).

Step 1: Understand the Trinomial Format

The trinomial "\(x^2 + 8xy + 15y^2\)" is a quadratic in the form \( ax^2 + bxy + cy^2 \). This format suggests that we will attempt to factor it into the product of two binomials.

Step 2: Identify Suitable Binomials

We need to identify two binomials of the form \((x + my)(x + ny)\) that multiply to give \(x^2 + 8xy + 15y^2\). Here, \(m\) and \(n\) must satisfy the equations: \(m + n = 8\) and \(m \cdot n = 15\).

Step 3: Solve for m and n

To factor correctly, we need two numbers that add up to 8 and multiply to 15. Testing combinations, we find that \(3 + 5 = 8\) and \(3 \times 5 = 15\). Thus, \(m = 3\) and \(n = 5\) are our solutions.

Step 4: Write the Factored Form

Using the values of \(m\) and \(n\) from Step 3, the trinomial can be factored as \((x + 3y)(x + 5y)\).

Step 5: Verify the Factorization

Expand \((x + 3y)(x + 5y)\) to ensure it equals the original trinomial: \((x + 3y)(x + 5y) = x^2 + 5xy + 3xy + 15y^2 = x^2 + 8xy + 15y^2\). The factorization is verified.

Key Concepts

Quadratic Equation

A quadratic equation is an important mathematical expression and often appears in the form of a trinomial, such as \( ax^2 + bxy + cy^2 \). It is characterized by the variable raised to the second power. In simple terms, it contains terms like \( x^2 \) with coefficients \( a, b, \) and \( c \). Quadratic equations can be recognized because they involve terms squared (hence the name 'quadratic', from the Latin word quadratus, which means square). Quadratic equations can represent various physical phenomena, from projectile motion to economic modeling. When dealing with quadratics, one common task is factoring, which means expressing the equation as a product of simpler expressions. This process often involves identifying what multipliers (or factors) can be used to rewrite the equation in a simpler form. This rewriting allows us to solve for the unknown variable by setting each factor to zero individually. Depending on the problem, these factors might represent real-world measurable quantities, and solving them can provide insights into the problem.

Binomial Factors

In mathematics, binomial factors are pairs of terms that multiply together to form a product. Essentially, a binomial is a polynomial with exactly two terms. For example, in the trinomial \( x^2 + 8xy + 15y^2 \), our goal is to break it into two binomial factors that look like \( (x + my)(x + ny) \). To solve this, we follow a systematic approach:

• Examine the middle term and the constant term in the trinomial.
• Find two numbers, \( m \) and \( n \), such that their sum is equal to the coefficient of the middle term (8 in this case), and their product is equal to the constant term (15).

This means finding combinations like \( m = 3 \) and \( n = 5 \), as these satisfy both conditions: \( 3 + 5 = 8 \) and \( 3 \times 5 = 15 \). Combining these, the factors of our trinomial \( x^2 + 8xy + 15y^2 \) become \( (x + 3y)(x + 5y) \). This factorization simplifies solving the equation and understanding the relationship between terms in polynomial expressions.

Polynomial Expressions

Polynomial expressions consist of terms that include variables raised to the power of whole numbers and are added, subtracted, or combined by multiplication. A polynomial expression like \( x^2 + 8xy + 15y^2 \) can be classified based on the number of terms it contains. If it has three terms, it is called a trinomial. The simplicity of a polynomial comes from its structure:

• Each term in a polynomial is composed of a coefficient (like 8 in 8xy) and variables raised to an exponent.
• Polynomials are sorted by the highest power of the variable, known as the degree (here, the degree is 2 due to \( x^2 \)).

When working on problems involving polynomials, factoring is a key technique to simplify computations and find solutions. Factoring, or rewriting a polynomial as a product of smaller polynomials or binomials, can reveal roots or simplify expressions into more manageable parts. Each polynomial expression tells a story of relationships represented by the terms, offering insight into the mathematical or real-world situation it describes.