Problem 36 - College Algebra with Modeling and Visualization

Question

Factor the expression completely. \(x^{2}-8 x+12\)

Step-by-Step Solution

Answer

The expression factors to \((x - 2)(x - 6)\).

Step 1: Understand the Quadratic Expression

The given expression is a quadratic trinomial of the form \( ax^2 + bx + c \). For this exercise, \( a = 1 \), \( b = -8 \), and \( c = 12 \). Our goal is to factor this expression into two binomials.

Step 2: Identify Potential Factor Pairs of c

We need two numbers that multiply to \( c = 12 \) and add up to \( b = -8 \). List the factor pairs of 12: (1, 12), (2, 6), (3, 4).

Step 3: Find the Correct Pair

From the pairs listed, identify the pair that adds up to -8. The correct pair is (-2, -6) because \((-2) \cdot (-6) = 12\) and \((-2) + (-6) = -8\).

Step 4: Write the Factors

Using the numbers found in Step 3, write the factors of the quadratic expression as: \((x - 2)(x - 6)\).

Step 5: Confirm the Factoring

To ensure the factors are correct, expand \((x - 2)(x - 6)\) to verify it results in the original expression. \((x - 2)(x - 6) = x^2 - 6x - 2x + 12 = x^2 - 8x + 12\). This confirms that the expression is factored correctly.

Key Concepts

Quadratic Trinomials

Quadratic trinomials are algebraic expressions composed of three terms where one of the terms includes a squared variable. Generally expressed in the form \( ax^2 + bx + c \), these terms are:

• \( ax^2 \) representing the quadratic term.
• \( bx \) the linear term, which involves just the variable \( x \).
• \( c \) the constant term, which is a number without a variable.

Understanding the structure of a quadratic trinomial is crucial because it guides how we approach factoring this kind of expression. Essentially, factoring involves reversing the process of expansion to rewrite the expression as a product of simpler 'binomial' expressions. This process usually starts by identifying a pair of numbers whose product and sum match the constant term (\( c \)) and the linear coefficient (\( b \)), respectively.
For instance, with the expression \( x^2 - 8x + 12 \), the trinomial's structure has \( a = 1 \), \( b = -8 \), and \( c = 12 \). By focusing on these values, we can break down and simplify this more complex expression into manageable parts.

Binomial Factors

Binomial factors are the building blocks into which quadratic trinomials are factored. Think of them as pairs of simpler expressions that, when multiplied together, reconstruct the original quadratic. The expression \( (x - 2)(x - 6) \) is an example of binomial factors derived from the quadratic trinomial \( x^2 - 8x + 12 \).

To identify the correct binomial factors:

• Look for numbers that multiply to the quadratic's constant term, which is \( c = 12 \) in our case.
• Also, these numbers should add up to the linear coefficient, \( b = -8 \).
• In this context, \(-2\) and \(-6\) fit the criteria: they multiply to \( 12 \) and add to \( -8 \).

Our final factored form of the quadratic \( x^2 - 8x + 12 \), written as \( (x - 2)(x - 6) \), reflects these considerations. This factorization simplifies solving equations involving this quadratic and offers insights into its roots, or solutions, where the expression equals zero.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations such as addition, subtraction, multiplication, and division. A quadratic trinomial like \( x^2 - 8x + 12 \) is one kind of algebraic expression that requires understanding specific operation rules to manipulate effectively.

Working with algebraic expressions involves:

• Recognizing the components (terms) and their relationships.
• Identifying operations that can simplify or transform the expression, such as factoring in the case of polynomials.
• Using equivalent expressions to make calculations easier, especially in problem-solving scenarios.

By mastering these skills, students can transition algebraic expressions from complex to simple forms, facilitating both computation and comprehension in mathematics. In the context of quadratic expressions, being able to factor them into binomial expressions is just one of the many abilities that showcase the power of working with algebraic expressions.