Problem 65 - Introductory Algebra for College Students

Question

Factor completely. $$x^{5}+3 x^{4} y-4 x^{3} y^{2}$$

Step-by-Step Solution

Answer

The factorized form of the polynomial is \(x^{3}(x^{2} + 3 x y - 4 y^{2})\)

Step 1: Identify the GCF

Identify the Greatest Common Factor (GCF) in all terms of the given equation. Here it is \(x^{3}\), since all terms include \(x\) raised to a power of at least 3.

Step 2: Factor out GCF

Factor out the GCF from all terms. If done correctly, this step will result in a simpler equation: \(x^{3}(x^{2} + 3 x y - 4 y^{2})\).

Step 3: Analyze the new expression

Now we will examine the polynomial within the parentheses. It is a quadratic polynomial which looks like a trinomial square but not exactly the same. So we cannot factorize it directly.

Step 4: Verifying Result

The result cannot be broken down further as there are no common factors in the inner bracket. So, the factorized form is \(x^{3}(x^{2} + 3 x y - 4 y^{2})\)

Key Concepts

Greatest Common Factor

When dealing with polynomial factorization, identifying the Greatest Common Factor (GCF) is a crucial first step. The GCF of algebraic expressions is the largest term that divides each term of the expression without leaving a remainder. In simpler terms, it's the biggest expression common to all parts of the polynomial.
Let's look at the polynomial \[ x^5 + 3x^4y - 4x^3y^2 \]
To find the GCF of these terms, observe the coefficients and variables. Here, all terms include a factor of \( x^3 \), as it is the smallest power of \( x \) common to all terms. By factoring out this GCF, the expression simplifies significantly to\[ x^3(x^2 + 3xy - 4y^2) \]
This step not only makes the polynomial smaller but also easier to work with for further factorization.

Quadratic Polynomial

A quadratic polynomial is a second-degree polynomial, often recognizable by its standard form \( ax^2 + bx + c \). In the context of the given exercise, once the GCF \( x^3 \) is factored out, what remains is the quadratic polynomial: \( x^2 + 3xy - 4y^2 \).
Quadratic polynomials have core characteristics:

• Three terms: a squared term, a linear term, and a constant.
• Can often be factored into two binomials.
• The solutions to the polynomial, or roots, can be found using various methods like the quadratic formula or factoring by grouping.

In this case, the polynomial resembles a trinomial square, but it doesn't exactly fit the perfect square form nor easily factor into binomials, suggesting that no further simple factorization is possible beyond its current state.

Algebraic Expressions

Algebraic expressions consist of variables, constants, and arithmetic operations like addition and multiplication. They are the building blocks of algebraic problems, and understanding their structure is key to solving them efficiently. An expression can be as simple as \( 5x \) or as complex as \( x^5 + 3x^4y - 4x^3y^2 \).
Key aspects of algebraic expressions include:

• Terms and operations that define the expression's structure.
• Variables which can be anything and constants that have fixed values.
• Operations like addition or multiplication indicate how the terms relate to one another.

In solving the exercise, identifying parts of the expression (like the common factor \( x^3 \)) simplifies the factorization process. Recognizing patterns and knowing how different terms interact are essential skills in working with algebraic expressions. This understanding empowers you to manipulate and simplify expressions like a pro, making complex algebraic challenges manageable.