Problem 11 - Algebra 1: Concepts and Skills

Question

Add or subtract, then factor and simplify. $$ \frac{10}{r^{2}+9 r+20}-\frac{-2 r}{r^{2}+9 r+20} $$

Step-by-Step Solution

Answer

The final answer is \( \frac{2}{r+4} \)

Step 1: Add the fractions

Since the fractions have the same denominator, we just add the numerators: \[ \frac{10 - (-2r)}{r^{2} + 9r + 20} \] which simplifies to \[ \frac{10 + 2r}{r^{2} + 9r + 20} \]

Step 2: Simplify the numerator

We can rearrange terms in the numerator, it looks like: \[ \frac{2r + 10}{r^{2} + 9r + 20} \]

Step 3: Factor the expressions

Factoring out a 2 from the numerator we get: \[ \frac{2(r+5)}{r^{2} + 9r + 20} \]. Factoring the denominator using the formula \( (x+a)(x+b) \) where a and b are roots of equation \(x^{2} + (a+b)x + ab = 0\), we get: \[ \frac{2(r+5)}{(r+5)(r+4)} \]

Step 4: Simplify the fraction

The \( (r+5) \) terms in the numerator and the denominator cancel each other out. \[ \frac{2}{r+4} \]

Key Concepts

Factoring Polynomials

Factoring polynomials is a key technique to simplify algebraic expressions, especially when dealing with rational expressions like fractions with polynomials in the numerator and denominator.
To factor a quadratic polynomial, such as \( r^2 + 9r + 20 \), look for two numbers that multiply to the constant term, 20, and add to the linear coefficient, 9. These numbers, called the roots of the polynomial, are 5 and 4 in this example. Thus, we can express the polynomial as \((r+5)(r+4)\).
This method of factoring is often called the reverse FOIL (First, Outer, Inner, Last) method, commonly used in algebra to decompose expressions. Factoring helps break down complex expressions into simpler, solvable parts.
Polynomials are factored by recognizing patterns, such as a difference of squares, sums, or differences of cubes, or by using formulas and trial methods.

Simplifying Expressions

Simplifying algebraic expressions involves combining like terms and reducing complex terms into simpler forms. This is crucial when dealing with large expressions.
For example, when simplifying \( \frac{10 - (-2r)}{r^2 + 9r + 20}\), combine the terms \(10 + 2r\) correctly. This removes any negative signs and combines coefficients correctly to make it simpler.
Next, look for common factors in the numerator and the denominator that can be canceled out. Factoring the numerator allows you to see \(2(r+5)\), illustrating a common factor with the denominator.
Simplification eliminates unnecessary parts to provide a cleaner, equivalent expression. Always check each step for common factors and correct signs to ensure accuracy.

Rational Expressions

Rational expressions are fractions where the numerator and/or the denominator are polynomials. Working with these requires ensuring they are in the simplest form.
Start by factoring both the numerator and the denominator, as seen in the expression \( \frac{2(r+5)}{(r+5)(r+4)} \). This reveals cancelable terms like \((r+5)\), which simplifies the expression further to \(\frac{2}{r+4}\).
Rational expressions can look intimidating, but by breaking them into smaller, factorable parts, the solution becomes straightforward. Ensuring both the numerator and denominator are factored is key to simplifying effectively.
Always watch out for potential restrictions in the domain, such as values that make the denominator zero (e.g., \(r eq -4\) in the final expression). This consideration is crucial for defining where the expression is valid.