Problem 21 - College Algebra Essentials

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 \( x^{2} - 8x + 15 \) is \( (x-5)(x-3) \).

Step 1: Identify the Values

The trinomial given in the equation is \( x^{2}-8 x+15 \) which is of the form \( ax^{2} + bx + c \). From the equation, we identify \( a = 1, b = -8, c = 15 \).

Step 2: Find the Factors

We need to find two numbers such that when multiplied, they give \( c = 15 \) and when added, they yield \( b = -8 \). The numbers are -5 and -3, because -5 times -3 equals 15 and -5 plus -3 equals -8.

Step 3: Factorize the Trinomial

Replace \( b \) with -5 and -3 in the trinomial equation \( x^{2} - 8x + 15 \). This gives us the factorized form of the trinomial as \( (x-5)(x-3) \).

Key Concepts

Quadratic equations

Quadratic equations are a fundamental part of algebra that involve expressions where the highest power of the variable is two. These equations generally take the form \( ax^2 + bx + c = 0 \). Here, \( a, b, \) and \( c \) are constants, and \( a eq 0 \). Solving a quadratic equation means finding the values of \( x \) that make the equation true. Quadratic equations can be solved using several methods including:

• Factoring
• Completing the square
• Using the quadratic formula

The method of factoring is often preferred due to its simplicity, especially when the equation is easily separable into two binomial expressions. In the exercise were tasked to factorize \( x^2 - 8x + 15 \), which is a typical quadratic equation in standard form.

Polynomial expressions

Polynomial expressions are mathematical expressions involving a sum of powers in one or more variables multiplied by coefficients. In simpler terms, these expressions contain terms like \( ax^n \) where \( n \) is a non-negative integer and \( a \) is a constant. A trinomial is a specific type of polynomial expression with exactly three terms. For example, \( x^2 - 8x + 15 \) is a trinomial because it includes three different terms: \( x^2 \), \(-8x\), and \( 15 \). When working with polynomial expressions, it's important to understand:

• The degree of the polynomial, which is the highest power of the variable present.
• The coefficients that impact the shape and position of the polynomial's graph.

In the context of the exercise, recognizing the trinomial form is crucial to applying factoring techniques effectively.

Algebraic factoring

Algebraic factoring is the process of breaking down a more complex expression into simpler parts, typically by finding two or more expressions whose product is the original. This is a vital skill in algebra and is particularly useful for simplifying polynomial expressions and solving quadratic equations.When factoring trinomials like \( x^2 - 8x + 15 \), the goal is to express it as a product of two binomials. The first step is to identify a pair of numbers that multiply to the constant term (15) and add up to the linear coefficient (-8), which are -5 and -3 in this case. The trinomial can then be expressed in factored form as \((x-5)(x-3)\).The benefits of algebraic factoring include:

• Simplifying complex expressions
• Solving equations more easily
• Finding roots or solutions to equations

With the aforementioned example, once the trinomial is factored, you can solve for the roots by setting each factor equal to zero, aiding in understanding the solutions of the quadratic equation.