Problem 18 - College Algebra

Question

Factor each trinomial, or state that the trinomial is prime. $$ x^{2}+8 x+15 $$

Step-by-Step Solution

Answer

The factorized form of the trinomial \( x^{2}+8 x+15 \) is \( (x+5)(x+3) \).

Step 1: Identify the coefficients

The coefficients for this trinomial are a=1, b=8, and c=15.

Step 2: Find the factors of 'c'

Find two numbers that both multiply to \( c=15 \) and add up to \( b=8 \). The numbers 5 and 3 meet these conditions because \( 5 \times 3 = 15 \) and \( 5 + 3 = 8 \).

Step 3: Factorize the quadratic trinomial

Substitute these numbers (5 and 3) into binomial terms yielding the factorization of the given trinomial as \( (x+5)(x+3) \).

Key Concepts

Quadratic Equations

Quadratic equations are mathematical expressions that involve a variable raised to the second power, hence the name "quadratic," derived from the Latin word for square. A typical quadratic equation looks like this: \[ ax^2 + bx + c = 0 \] where:\

• \( a \), \( b \), and \( c \) are constants
• \( a \) ≠ 0, as it defines the equation being quadratic
• \( x \) is the unknown variable

These equations are fundamental in algebra and appear in problems ranging from physics to finance. Understanding the structure of a quadratic equation is key before tackling its solutions, whether by factoring, completing the square, or using the quadratic formula. Let's dive into identifying the core elements, which is the first step in the factorization process.

Coefficient Identification

Identifying coefficients is crucial when working with quadratic equations. Each quadratic equation is characterized by three essential parts, the coefficients that define its shape and solutions. Here's what you need to know:

• Leading Coefficient (a): This is the coefficient of the square term \( x^2 \) and indicates the parabola's width and direction - up if positive, down if negative.
• Linear Coefficient (b): Positioned with the \( x \) term, it affects the parabola's symmetry and placement relative to the y-axis.
• Constant Term (c): This number stands alone, determining where the parabola intersects the y-axis.

For example, in the trinomial \( x^2 + 8x + 15 \), \( a = 1 \), \( b = 8 \), and \( c = 15 \). Recognizing these parts allows you to proceed to the next steps in solving and factoring the equation correctly.

Polynomial Factorization

Polynomial factorization transforms a polynomial into simpler multiples or products. This technique is vital for simplifying expressions and solving equations. For a quadratic trinomial like \( x^2 + 8x + 15 \), factorization involves rewriting it as the product of two binomials: \[ (x + 5)(x + 3) \] To factor trinomials effectively, follow these general steps:

• Identify the coefficients \( a \), \( b \), and \( c \).
• Find two numbers that multiply to \( c \) while adding to \( b \).
• Use these numbers to separate and break down the middle term before grouping terms to further simplify into a binomial product.

This method leverages the ability to "see" numbers abstractly and manipulate them, emphasizing the importance of practice in becoming proficient at this algebraic skill.

Binomials

Binomials are a fundamental component in algebra, consisting of two terms connected by either addition or subtraction. They appear frequently when dealing with expressions and equations, especially in factorization. The expression \( (x + 5)(x + 3) \) is a perfect example of a factored trinomial into a product of two binomials. Expanding this product using the distributive property will return you back to the original trinomial \( x^2 + 8x + 15 \). When writing binomials, the order and sign of the terms are crucial. For instance, \((x + 3)(x + 5)\) and \((x - 3)(x - 5)\) produce different results. Recognizing binomial patterns is foundational to simplifying algebraic expressions and especially helpful for solving quadratic equations efficiently.