Problem 13 - Introductory Algebra for College Students

Question

Use the method of your choice to factor each trinomial, or state that the trinomial is prime. Check each factorization using FOIL multiplication. $$3 x^{2}-22 x+7$$

Step-by-Step Solution

Answer

The factorization of the given trinomial \(3x^{2}-22x+7\) is \((3x - 1)(x - 7)\).

Step 1: Identify the structure

The given trinomial \(3x^{2}-22x+7\) has the structure of a quadratic trinomial \(ax^{2} + bx + c\), where a=3, b=-22, c=7.

Step 2: Factorise the trinomial

Factorising involves finding two numbers that multiply to a*c (i.e., 3*7= 21) and add to give b (i.e., -22). The numbers that satisfy these conditions are -21 and -1. Therefore, the trinomial can be written as \(3x^{2}-21x-x+7\) . Grouping the four terms into two groups results in \((3x^{2}-21x) + (-x+7)= 3x*(x-7)-1*(x-7)\) . Finally, factor out the common terms to get \((3x - 1)(x - 7)\) .

Step 3: Check the factorisation using FOIL multiplication

The check involves applying the FOIL method to the factored form \((3x - 1)(x - 7)\) to ensure that it yields the original trinomial. FOILing gives: First: \(3x*x=3x^{2} ,\) Outside: \(3x*-7 = -21x, \) Inside: \(-1*x = -x, \) Last: \(-1*-7 = 7. \) Adding these together, \( 3x^{2} -21x -x + 7 = 3x^{2} -22x + 7, \) which is the original trinomial.

Key Concepts

Quadratic Equation

A quadratic equation is a type of polynomial equation with the highest exponent of the variable being 2. These equations take the general form of \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants. In this form, \( a \) is not equal to zero because if it were, the equation would not be quadratic but linear instead.

When dealing with quadratic equations, it's important to consider the arrangement of the equation, known as the standard form. For example, the expression given in the exercise \( 3x^2 - 22x + 7 \) fits the standard form, where \( a = 3 \), \( b = -22 \), and \( c = 7 \).

To solve these equations, different methods like factoring, completing the square, and using the quadratic formula are commonly employed. In this exercise, the focus is on factoring to simplify the problem into smaller linear components.

FOIL Method

The FOIL method is a technique used for multiplying two binomials. FOIL stands for First, Outside, Inside, Last, representing the order in which you multiply the terms in each binomial.

For example, consider two binomials \((a + b)(c + d)\). When applying the FOIL method, you perform the following steps:

• First: Multiply the first terms of each binomial: \(a \cdot c\).
• Outside: Multiply the outer terms: \(a \cdot d\).
• Inside: Multiply the inner terms: \(b \cdot c\).
• Last: Multiply the last terms of each binomial: \(b \cdot d\).

After applying FOIL, you add the resulting products together.

In the solution provided, the FOIL method is used to verify the factorization of \((3x - 1)(x - 7)\). By multiplying and then summing up these products, one confirms that the factorization correctly reconstructs the original trinomial.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables, and operations. They do not include an equality sign, unlike equations. In algebra, expressions help in forming equations and inequalities. Understanding how to manipulate these expressions is crucial, as they form the basis for solving problems in algebra.

In the trinomials we deal with, each term is part of an algebraic expression. For instance, the trinomial \(3x^2 - 22x + 7\) is an algebraic expression with three terms. Learning to identify parts of these expressions allows us to execute more complex operations, such as factoring, which simplifies the terms into useful forms. This transformation is critical for solving algebraic equations efficiently.

Recognizing these patterns in algebraic expressions allows you to break them down, factor them, and solve equations more readily. This becomes especially important with higher-level algebra, where managing expressions effectively leads to the successful solving of equations.