Problem 50 - Beginning and Intermediate Algebra

Question

Factor completely. Check your answer. $$r^{2}-9 r s+20 s^{2}$$

Step-by-Step Solution

Answer

The quadratic expression \(r^2 - 9rs + 20s^2\) can be factored completely as \((r - 4s)(r - 5s)\).

Step 1: Identify the coefficients and constant term

We are given the quadratic expression \(r^2 - 9rs + 20s^2\). In this expression, the coefficient of the quadratic term \(r^2\) is 1, the coefficient of the linear term \(-9rs\) is -9, and the constant term is \(20s^2\).

Step 2: Find two numbers that multiply to the product of leading coefficient and constant term, and add up to the linear term's coefficient

We need to find two numbers that multiply to give 1 * 20s², which is 20s², and add up to -9. The two numbers are -4 and -5 because (-4) * (-5) = 20, and (-4) + (-5) = -9.

Step 3: Rewrite the expression using the two numbers as the coefficients of the linear terms

Rewrite the middle term, -9rs, as the sum of -4rs and -5rs. This results in the following expression: \(r^2 - 4rs - 5rs + 20s^2\).

Step 4: Factor the expression by grouping

Group the first two terms and the last two terms: \((r^2 - 4rs) + (-5rs + 20s^2)\). Factor the common terms from each group: \(r(r - 4s) - 5s(r - 4s)\).

Step 5: Factor the common binomial

Notice that both terms have a common binomial factor of (r - 4s). Factor this binomial out of the expression: \((r-4s)(r-5s)\).

Step 6: Check the answer

To check the answer, expand the factored form and see if it matches the original expression: \((r-4s)(r-5s) = r^2 - 5sr - 4sr + 20s^2 = r^2 - 9rs + 20s^2\) Since the expanded form matches the original expression, the factored form is correct. So, \(r^2 - 9rs + 20s^2\) can be factored completely as \((r - 4s)(r - 5s)\).

Key Concepts

Quadratic Expression

A quadratic expression is a type of polynomial that typically looks like a trinomial in the form \( ax^2 + bx + c \), where \( a, b, \) and \( c \) are constants. In the given exercise, the quadratic expression is \( r^2 - 9rs + 20s^2 \).
This expression involves two variables, \( r \) and \( s \), making it a bivariate quadratic polynomial. It’s called "quadratic" because

1. The highest power is 2, residing in \( r^2 \).
2. It has three terms, making it a trinomial.

Recognizing the structure of a quadratic expression is crucial for the method of factoring to simplify or solve equations involving these expressions.

Factoring by Grouping

Factoring by grouping is an algebraic method used to factor polynomials. The process involves grouping terms to reveal common factors, which makes the expression easier to factor. In the solution for \( r^2 - 9rs + 20s^2 \), we identified the need to group by considering terms that can be paired for factoring.
The method consists of the following steps:

• Split the middle term into two terms that add up to it, given the conditions of multiplication and addition of pairs.
• Group terms in pairs, typically two sets.
• Factor out the greatest common factor from each group.
• Identify and extract the common binomial factor, if present, to achieve a factored expression.

This method is especially useful when dealing with polynomials where the coefficient of the squared variable is 1, aiding in clear simplification.

Polynomial Coefficients

In algebra, polynomial coefficients are the numbers that multiply the terms in a polynomial. Understanding coefficients is key to the structure and behavior of these expressions. For the quadratic expression \( r^2 - 9rs + 20s^2 \), the coefficients are:

• Leading coefficient of \( r^2 \): 1
• Coefficient of \( -9rs \): -9
• Constant term coefficient \( 20s^2 \): 20 (though this appears as part of the binomial \((s^2)\))

These coefficients play a vital role in determining the expression's shape and direction when graphed.
They also help us in the process of factoring by conveying relationships between terms, such as their sum and product, which are key in methods like factoring by grouping.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain numbers, variables and operations. It is a general term that encompasses all sorts of combinations, from simple monomials to complex polynomials. The expression \( r^2 - 9rs + 20s^2 \) is an example of a multivariable algebraic expression.
Such expressions can be processed to simplify equations or solve for variable values, often by factoring, expansion, or substitution techniques. Understanding and manipulating algebraic expressions is foundational in algebra, enabling solutions to equations that model real-world situations. Through methods like factoring by grouping, you can simplify these expressions into a product, like turning \( r^2 - 9rs + 20s^2 \) into \((r - 4s)(r - 5s)\), which can then be easier to graph, interpret, or solve.