Problem 21 - Beginning and Intermediate Algebra

Question

Divide. $$\frac{g^{2}+9 g+20}{g+5}$$

Step-by-Step Solution

Answer

The simplified form of the given rational expression \(\frac{g^2 + 9g + 20}{g+5}\) is \(g+4\), after factoring the numerator and canceling out the common factors.

Step 1: Factor the numerator

We must first factor the numerator polynomial, \(g^2 + 9g + 20\). Look for two numbers that multiply to 20 and add to 9. Numbers that multiply to 20: (1, 20), (2, 10), (4, 5) The numbers 4 and 5 both add to 9, therefore, the factorization of the given polynomial is: \((g+4)(g+5)\)

Step 2: Rewrite the expression

Now that we have factored the numerator, rewrite the entire rational expression: $$\frac{g^{2}+9 g+20}{g+5} = \frac{(g+4)(g+5)}{g+5}$$

Step 3: Cancel out common factors

As we can see, both the numerator and denominator share a common factor of \((g+5)\). We can now cancel out this factor: $$\frac{(g+4)(g+5)}{g+5} = g+4$$

Step 4: Write the simplified expression

The simplified expression is the remaining term after cancelling out the common factor: $$g+4$$ The rational expression is now simplified as \(g+4\).

Key Concepts

Factoring Polynomials

When it comes to factoring polynomials, the main aim is to break down a polynomial into simpler products of more manageable polynomials. Think of it as trying to pick apart a complex-looking algebraic expression into its building blocks.
To factor a quadratic polynomial like \(g^2 + 9g + 20\), we begin by looking for two numbers that multiply together to give the constant term (20), and at the same time, add up to the coefficient of the middle term (9).

Here is how we can approach it:

• List all factor pairs of the constant term: (1, 20), (2, 10), (4, 5).
• Identify which pairs add up to the middle coefficient: the pair (4, 5) works since 4 + 5 = 9.
• Rewrite the polynomial as \(g^2 + 9g + 20 = (g + 4)(g + 5)\).

By factoring, we simplify the division process in polynomial division significantly.

Rational Expressions

Rational expressions are fractions where both the numerator and the denominator are polynomials. In this division problem, \(\frac{g^2 + 9g + 20}{g+5}\) is a rational expression.

Understanding rational expressions involves recognizing:

• They simplify by canceling common factors in the numerator and denominator.
• They must be factored, if possible, to reveal these common factors.

A crucial aspect of working with rational expressions is ensuring the expressions are entirely in their simplest form before making calculations. This involves factoring first and then reducing conspicuously as shown in the result, \(\frac{(g+4)(g+5)}{g+5}\). By removing the common factor \(g+5\), the rational expression simplifies into a form that's easy to interpret.

Simplifying Expressions

In mathematics, simplifying expressions is all about making them easier to work with. To simplify a rational expression, crucial steps follow the strategy of recognizing and eliminating unnecessary complexity.
In the given solution, the expression is simplified by canceling the factor \(g+5\) from both the numerator and the denominator.
Here's why it works:

• Factoring reveals common elements, like a puzzle showing hidden pieces.
• Canceling is like knocking away identical parts of the puzzle that don't need to be duplicated.

By canceling out \(g+5\), what remains, \(g+4\), is a streamlined and simplified version of the expression originally given. This simplification often makes further calculations or solving problems much more straightforward, allowing focus on key components without unnecessary clutter.