Problem 68 - College Algebra and Calculus: An Applied Approach

Question

Simplify the rational expression. $$\frac{x^{3}+15 x^{2}+68 x+96}{x+4}$$

Step-by-Step Solution

Answer

\((x+8)(x+3)\

Step 1: Factorizing the numerator

Firstly, identify that the cubic polynomial equation in the numerator is actually factorable. So, let's start by factoring the cubic polynomial \(x^{3}+15x^{2}+68x+96\) by factor by grouping. The first group is \(x^3+15x^2\) and the second group is \(68x+96\). Factor out the GCD, \(x^2\) from the first group and 4 from the second group. This gives \(x^2(x+15)+4(17x+24)\). Now, continue to factor so that the terms may be expressed as the product \((x+8)(x^2+7x+12)\)

Step 2: Factorize Further

Factor further by breaking down the quadratic polynomial. It can be factored as \((x+8)(x+4)(x+3)\)

Step 3: Cancel Out Common Factors

The rational expression is now \(\frac{(x+8)(x+4)(x+3)}{x+4}\). Notice that \(x+4\) appears in both the numerator and denominator. So, these can cancel out, leaving the simplified expression as \(x^{2}+11x+24\)

Step 4: Simplify Further

The expression \(x^{2}+11x+24\) can be further simplified as \((x+8)(x+3)\). So, the rational expression simplifies to \((x+8)(x+3)\). This is the final simplification of the given expression

Key Concepts

Factorization

Factorization is a method used to break down complex expressions into simpler components, called factors. In the context of polynomials, factorization can help us to simplify expressions and solve equations more easily. To factor a polynomial, you need to identify its greatest common divisor (GCD) or try other methods like grouping.

• In the original example, we started with the polynomial: \(x^3 + 15x^2 + 68x + 96\).
• We used factor by grouping, separating the polynomial into groups: \((x^3 + 15x^2)\) and \((68x + 96)\).
• The GCD was factored out from each group, yielding \(x^2(x + 15) + 4(17x + 24)\).
• This process transforms a complex expression into a more manageable one, with the polynomial eventually being factorized as \((x + 8)(x^2 + 7x + 12)\).

Breaking down complex polynomials into factors makes it easier to work with them in algebraic operations like division and simplification.

Polynomial Division

Polynomial division is a method used to divide one polynomial by another. This is similar to long division with numbers but involves variables and coefficients. In this exercise, we performed polynomial division to simplify the rational expression.

• Once the numerator was fully factored into \((x + 8)(x + 4)(x + 3)\), it was easier to see common factors with the denominator \((x + 4)\).
• We cancel the common factor \((x + 4)\) from the numerator and the denominator.
• This leaves us with a simpler polynomial expression: \((x + 8)(x + 3)\).

Understanding polynomial division helps you simplify rational expressions by identifying and canceling out common factors. This process is crucial for reducing expressions in algebraic problems.

Simplification

Simplification involves reducing expressions to their most basic form. In this exercise, we started with a complex rational expression and simplified it through factorization and polynomial division.

• After canceling out the common factors, the rational expression \(\frac{(x + 8)(x + 4)(x + 3)}{x + 4}\) simplifies to \((x + 8)(x + 3)\).
• This indicates the expression has been fully simplified, as it cannot be reduced further without altering its value.
• Each step in simplification helps clarify the structure of the expression, revealing any inherent patterns or symmetries.

Simplification is an essential skill in algebra, allowing you to condense expressions into more approachable forms, making them easier to solve or manipulate.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations like addition, subtraction, multiplication, and division. They form the foundation of many mathematical tasks, including solving equations and simplifying expressions.

• The original problem represented an algebraic expression in the form of a rational expression: \(\frac{x^3 + 15x^2 + 68x + 96}{x + 4}\).
• The goal was to simplify this expression by identifying and removing common factors through factorization.
• Being comfortable with algebraic expressions allows you to manipulate and solve a range of mathematical problems efficiently.

Understanding and working with algebraic expressions equip you with the tools needed to tackle advanced mathematical concepts and real-world problems.