Problem 21 - College Algebra

Question

In Exercises \(17-30,\) factor each trinomial, or state that the trinomial is prime. $$x^{2}-8 x+15$$

Step-by-Step Solution

Answer

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

Step 1: Identify the coefficients

The given trinomial is \(x^{2} - 8x + 15\). Here, the coefficient of \(x^{2} (a)\) is '1', the coefficient of \(x (b)\) is '-8', and the constant term \(c\) is '15'.

Step 2: Find the factors of the constant term

Find two numbers that multiply to give \(c (15)\) and add up to give \(b (-8)\). These numbers are '-3' and '-5', because \(-3 \times -5 = 15\) and \(-3 + -5 = -8\).

Step 3: Write down the factors

Split the middle term \(-8x\) in terms of the found factors and rewrite the trinomial as \(x^{2} - 3x - 5x + 15\).

Step 4: Use the distributive property to factor

Factor by grouping the terms in pairs: \(x(x - 3) - 5(x - 3)\). Here, \(x - 3\) is the common factor.

Step 5: Write the final factored form

Write the final factored form as: \((x - 3)(x - 5)\)

Key Concepts

Trinomial Expression

A trinomial expression is a type of algebraic expression that contains three terms. In the context of polynomials, a trinomial typically takes the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(x\) is a variable.
For instance, in our exercise, the expression \(x^2 - 8x + 15\) is a trinomial because it consists of three distinct terms: the quadratic term \(x^2\), the linear term \(-8x\), and the constant term \(15\).
The structure of trinomial expressions allows us to apply specific mathematical techniques, such as factoring, to simplify or solve them. Understanding each part of the trinomial helps in recognizing patterns that can lead to more efficient solutions.

Polynomial Factorization

Polynomial factorization involves breaking down a polynomial into a product of simpler polynomials. This process is crucial in simplifying polynomials and finding their roots.
To factor trinomials, especially those in the form \(ax^2 + bx + c\), the goal is to find two binomials \((px + q)(rx + s)\) such that the product equals the original trinomial.
When we have a trinomial with a leading coefficient of 1, like \(x^2 - 8x + 15\), we can focus on the constant term \(c\) and the linear coefficient \(b\). Our task is to find two numbers that multiply to \(c\) and add up to \(b\) (e.g., \(-3\) and \(-5\) for this trinomial). This approach allows us to break down the trinomial into factors efficiently.

Algebraic Expressions

Algebraic expressions combine numbers and variables with operations like addition, subtraction, multiplication, and division. These expressions can range in complexity from simple monomials to more intricate polynomials, like trinomials.
Working with algebraic expressions often requires simplification to solve equations or inequalities. Understanding the properties and behavior of these expressions is foundational in algebra. For instance, recognizing when to factor or distribute terms can greatly simplify problem-solving.
In the exercise provided, we factored the algebraic expression \(x^2 - 8x + 15\), demonstrating that it can be rewritten as a product of two simpler expressions, \((x - 3)(x - 5)\). With practice, mastering the manipulation of algebraic expressions becomes an invaluable tool in various mathematical disciplines.