Problem 45 - Elementary and Intermediate Algebra

Question

$$a^{2}-5 a b+6 b^{2}$$

Step-by-Step Solution

Answer

The expression factors to \((a - 3b)(a - 2b)\).

Step 1: Identify the Terms

The expression is given as \(a^{2} - 5ab + 6b^{2}\). It is a trinomial with three terms that contain the variables \(a\) and \(b\).

Step 2: Analyze Possible Factorization

Consider the expression as a quadratic in terms of \(a\), which gives a leading coefficient of 1. The expression is an attempt to be factorized as \((a - x)(a - y)\).

Step 3: Understand Factoring Requirements

To factor \(a^{2} - 5ab + 6b^{2}\), we need two numbers whose product is 6 and sum is -5. Think of \(6b^{2}\) as \((3b)(2b)\) and find appropriate factors.

Step 4: Find Suitable Factors

Match up the numbers that multiply to 6 \((3b \times 2b)\) and that sum up to -5 \(-3b - 2b\). These are \(-3b\) and \(-2b\).

Step 5: Factor the Expression

Write the expression \(a^2 - 5ab + 6b^2\) in terms of these factors: \( (a - 3b)(a - 2b)\). Therefore, the expression can be factored as \((a - 3b)(a - 2b)\).

Step 6: Verify the Solution

Expand \((a - 3b)(a - 2b)\) to check if it equals the original expression. Multiply: \[(a - 3b)(a - 2b) = a^{2} - 2ab - 3ab + 6b^2 = a^2 - 5ab + 6b^2\]. The factored form is correct.

Key Concepts

Trinomial Expressions

Trinomial expressions are a type of polynomial that includes exactly three terms. These terms are usually separated by a plus or minus sign. In the given expression, \(a^{2} - 5ab + 6b^{2}\), each term consists of different combinations of the variables \(a\) and \(b\). Specifically, it features:

• First term: \(a^{2}\) - The square of \(a\).
• Second term: \(-5ab\) - This term involves both \(a\) and \(b\) and combines them linearly.
• Third term: \(6b^{2}\) - The square of \(b\) multiplied by 6.

Understanding these components is crucial when factoring the expression, as each term interacts to influence the overall structure. Analyzing the interaction between these terms, mainly focusing on coefficients and powers, forms the backbone of operations performed on the trinomial.

Polynomial Factorization

Polynomial factorization is the process of breaking down a polynomial into simpler 'factors' that, when multiplied together, yield the original polynomial. This is similar to breaking down numbers into prime factors but done in an algebraic context. When handling trinomial expressions like \(a^{2} - 5ab + 6b^{2}\), we often aim to transform the expression into a product of binomials.
Consider the target expression to be factored:

• Identify possible binomial factors: We need to determine two binomials that multiply to give the trinomial.
• Find pairs of numbers: For terms other than \(a^{2}\), find numbers that satisfy specific multiplication and addition properties related to the trinomial's coefficients.
• Utilize trial and error: Sometimes, it's necessary to test different combinations until the multiplication and addition properties fit perfectly.

In the given exercise, the expression was factored into \((a - 3b)(a - 2b)\), which on expansion provides the original polynomial. This demonstrates a perfectly executed polynomial factorization.

Algebraic Expressions

Algebraic expressions are combinations of letters and numbers where letters represent variables. They are used to represent general or specific relationships between quantities.
In the expression \(a^{2} - 5ab + 6b^{2}\), we have two variables, \(a\) and \(b\). These expressions form the foundation of algebra, allowing for the representation of general relationships that can be manipulated to solve problems.
Key characteristics of algebraic expressions include:

• Coefficients: Numbers multiplying the variables, such as -5 in \(-5ab\).
• Variables: Letters like \(a\) and \(b\) can take different numerical values, depending on the context.
• Constants: Numbers on their own, if present, but absent in this specific trinomial.

Manipulating these elements through operations like factoring, expanding, or substituting, empowers one to discover more about the relationships within mathematical problems, particularly those involving algebra.