Problem 19 - College Algebra

Question

In Exercises \(17-30,\) factor each trinomial, or state that the trinomial is prime. $$x^{2}-2 x-15$$

Step-by-Step Solution

Answer

The factored form of the trinomial \(x^{2}-2 x-15\) is \((x-5)(x+3)\)

Step 1: Identify the coefficients

In the given quadratic trinomial \(x^{2}-2 x-15\), the coefficient of \(x^{2}\) is \(1\), the coefficient of \(x\) is \(-2\), and the constant term is \(-15\).

Step 2: Find the factors

We now need to find the factors of \(-15\) that add up to \(-2\). The factors of \(-15\) are \(1, -1, 3, -3, 5, -5, 15, -15\). The pair of factors that add up to \(-2\) is \(3\) and \(-5\).

Step 3: Write the factors

We are now ready to write our quadratic trinomial in its factored form. Use the factors to write two binomials. Our trinomial factors into \((x-5)(x+3)\).

Key Concepts

Understanding Trinomial Expressions

A trinomial expression is a type of polynomial that consists of exactly three terms. These terms can include variables and constants, and they can appear in any order. For example, in the expression \(x^2 - 2x - 15\), the terms are:

• \(x^2\) - this is the quadratic term because it includes the square of a variable;
• \(-2x\) - this is the linear term because it includes the variable to the first power;
• \(-15\) - this is the constant term as it does not have any variables.

Trinomials can be part of larger mathematical expressions, and they often play an essential role in solving equations and modeling real-world scenarios. Recognizing the components of a trinomial is crucial for successfully factoring it, especially when preparing to solve quadratic equations or other advanced algebraic problems.

Quadratic Equations Explained

Quadratic equations are special types of equations where the highest variable exponent is 2, which means they have a square term. These equations are in the general form \(ax^2 + bx + c = 0\), with \(a\), \(b\), and \(c\) being coefficients or constants. In our specific example, \(x^2 - 2x - 15 = 0\), we can identify:

• \(a = 1\) - the coefficient of the \(x^2\) term;
• \(b = -2\) - the coefficient of the \(x\) term;
• \(c = -15\) - the constant term.

Solving quadratic equations often involves finding the values of \(x\) that make the equation true. These values are called solutions or roots. There are several methods to solve quadratic equations, including factoring, using the quadratic formula, or completing the square. Factoring is typically the first approach when dealing with simple quadratics, as it can be the most straightforward and intuitive method.

The Factorization Process

Factoring polynomials is breaking them down into simpler terms or expressions that can be easily multiplied to get the original polynomial. This process is crucial in solving equations and simplifying expressions. When factoring a quadratic trinomial like \(x^2 - 2x - 15\), follow these steps:- **Identify Coefficients:** Recognize the coefficients in the trinomial (\(a=1\), \(b=-2\), \(c=-15\)).- **Find Suitable Numbers:** Look for two numbers that multiply to \(ac\) (the product of the first and last coefficients, which is \(-15\) here) and add to \(b\) (which is \(-2\)).- **Write the Factors:** These numbers help break down the middle term, writing the trinomial as \((x - 5)(x + 3)\).This specific factorization reveals the roots of the equation when you set each binomial factor equal to zero. Factoring not only simplifies expressions but also reveals key properties of the functions they represent.