Question

Simplify the expression, if possible. $$ \frac{x^2+11 x+18}{x^3+8} $$

Short answer

The simplified expression is \( \frac{(x+2)(x+9)}{x^3 + 8} \)

Step-by-step solution

Factorize the numerator

The first step is to factor the numerator. By finding two numbers that multiply to 18 and add to 11, we see that these numbers are 2 and 9. Therefore, \(x^2+11x+18\) can be written as \( (x+2)(x+9) \)

Analyze the denominator

The denominator of the fraction is \(x^3 + 8\). This is already in a simplified form. We also look for common factors in the numerator and denominator but there are none.

Write the final expression

After factoring the numerator and analyzing the denominator, we just have to write down the final factored form of the function. The final expression is \(\frac{(x+2)(x+9)}{x^3 + 8}\)

Key concepts

Factoring Polynomials

Factoring polynomials is like finding two or more numbers that can be multiplied to get the original number. When you factor polynomials, you're doing this with terms made up of variables and coefficients.
To factor a polynomial such as \(x^2 + 11x + 18\), you need to find two numbers that can multiply to the constant term (18) and add up to the linear coefficient (11). In this case, the numbers are 2 and 9. Therefore, the expression \(x^2 + 11x + 18\) can be rewritten as \((x + 2)(x + 9)\).
A helpful tip for factoring is to always consider simple methods first, like looking for a common factor before applying more complex techniques like grouping or using the quadratic formula.

Rational Expressions

Rational expressions are similar to fractions, but instead of whole numbers, they involve polynomials. They have a numerator and a denominator, both of which are polynomials. For example, consider \(\frac{x^2+11x+18}{x^3+8}\).
Just like with fractions, you can perform operations like addition, subtraction, multiplication, and division on rational expressions. The key is to look for ways to simplify them by factoring and canceling out common terms.
However, it's important to check your work because you can't divide by zero, so make sure that the final expression's denominator doesn't equal zero. This ensures that the expressions are valid across all values except those making the denominator zero.

Simplifying Expressions

Simplifying expressions involves rewriting them in a more compact or manageable form without changing their value. The primary goal is to reduce the complexity of the expression for easier manipulation or understanding.
In our example, simplifying starts with factoring the numerator \(x^2+11x+18\) to \((x+2)(x+9)\). The denominator, \(x^3+8\), already appears in its simplest form. Thus, the expression is expressed as \(\frac{(x+2)(x+9)}{x^3+8}\).
When simplifying:

• Factor whenever possible to reduce terms.
• Cancel any common factors between the numerator and the denominator.
• Check to ensure no restrictions arise, like zero in denominators that might make the expression undefined.

These steps should help you simplify most polynomial expressions effectively and efficiently.