Problem 72 - A Graphical Approach to College Algebra

Question

Factor by any method. $$8 r^{2}-3 r s+10 s^{2}$$

Step-by-Step Solution

Answer

Cannot be factored over integers; try over complex numbers.

Step 1: Recognize the Problem Type

This problem requires factoring a quadratic polynomial in two variables, specifically in the form \( ax^2 + bxy + cy^2 \). Here, it's \(8r^2 - 3rs + 10s^2\).

Step 2: Check the Quadratic Formula

For quadratics in two variables like \( ax^2 + bxy + cy^2 \), direct factoring is difficult. Note that formulas such as completing the square or using specific factorization methods can help.

Step 3: Attempt Factoring by Grouping

Grouping involves rearranging and factoring terms, but the given quadratic doesn't lend itself easily to this due to distinct variable terms \(rs\). Typically, you would rearrange into terms that share common factors, but this isn't straightforward here.

Step 4: Use the AC Method

The AC method suggests finding two numbers that multiply to \( a \times c \) (80 here) and add to \( b \) (-3 here). Let's find two such numbers.

Step 5: Evaluate Combinations

Try combinations of numbers to get product 80 and sum -3, e.g., \(10\) and \(-8\). However, no integer combinations work due to coefficient constraints and result in complications.

Step 6: Determine Non-Factorability Over Integers

Since integer combinations don't work effectively, the polynomial \( 8r^2 - 3rs + 10s^2 \) cannot be factored neatly over the integers, sticking to the sums-product relation of factoring.

Step 7: Consider Factorization Over Complex Numbers

For advanced cases, if allowed, consider factoring over complex numbers. However, typical high school problems expect integer or real solutions.

Key Concepts

Polynomial Expressions

Polynomial expressions are mathematical statements that can include constants, variables, and exponents combined using operations like addition, subtraction, multiplication, and sometimes division. These expressions can have multiple terms, and when dealing with polynomials, one often seeks to simplify or rearrange them for easier handling. A polynomial is classified by its degree, which reflects the highest power of the variable within the expression. Understanding the basic structure of polynomials is key in manipulating and solving them effectively. In the context of factoring, polynomials are broken down into products of simpler polynomials. Different methods such as factoring by grouping, using special products, or the quadratic formula can be employed depending on the expression type and structure. Being adept at recognizing polynomial forms helps make the factoring process quicker and more intuitive, as seen with expressions like \(8r^2 - 3rs + 10s^2\). This expression is quadratic with respect to both variables, setting the stage for more advanced factoring techniques.

Two Variables Quadratic

A two variables quadratic polynomial, like the expression \(8r^2 - 3rs + 10s^2\), involves terms of two different variables mixed together. The general form for such polynomials is \(ax^2 + bxy + cy^2\). These are more complex than single-variable quadratics and require careful attention to the relations between terms. The presence of the cross term \(bxy\), like \(-3rs\) in this example, adds an extra layer of complexity because it ties the two variables together. This pairing of terms complicates classical methods of factorization like basic grouping, which rely on finding common factors easily. For two variables quadratics, examining the coefficients, especially the interactions between the "mixed" terms, is important. While it's possible to factor these types of expressions using methods like trial and error or specialized formulas, it often involves more advanced strategies or the use of algebraic software for confirmation.

AC Method

The AC Method is a systematic approach to factoring quadratic expressions, particularly helpful when direct methods are cumbersome. It's named so because it involves multiplying the coefficients of the first and the last term of the quadratic expression. For instance, in \(8r^2 - 3rs + 10s^2\), the AC method requires you to multiply \(a\) (8) and \(c\) (10), finding values that multiply to this product (80) and add up to the middle coefficient \(b\) (-3). This factor pair helps you break apart the middle term, aiding its rearrangement into factorable groups. Despite the reliability of the AC Method, it isn't foolproof. Some quadratic expressions, especially those involving multiple variables or peculiar coefficients, may resist clean factorization over integers, as seen with the exercise provided. Understanding the limitations of this method helps set realistic expectations and guides you towards alternative strategies, such as exploring complex numbers for a broader range of solutions.