Problem 140 - College Algebra Essentials

Question

$$\text { factor completely.}$$ $$x^{4}-y^{4}-2 x^{3} y+2 x y^{3}$$

Step-by-Step Solution

Answer

The factored form of \(x^{4}-y^{4}-2 x^{3} y+2 x y^{3}\) is \(x^3(x - 2y) - y^3(2x - y)\).

Step 1: Group the expression

Firstly, regroup the given expression \(x^{4}-y^{4}-2 x^{3} y+2 x y^{3}\) as \((x^4 - 2x^3y )+ (2xy^3 - y^4)\). This will allow us to factorize using the identity \(a^2 - b^2 = (a-b)(a+b)\)

Step 2: Factorize the first term

Now, factorize the first group. It can be rewritten as \(x^3(x - 2y)\). Here, \(x^3\) is taken as a common factor from the two terms.

Step 3: Factorize the second term

Similarly, the second group can be rewritten as \(-y^3(2x - y)\). Here, \(-y^3\) is common in both the terms and has been taken out.

Step 4: Rewrite the expression

Combine the results of step 2 and step 3 to get the factorized form of the complete expression as \((x^3 - y^3)(x - 2y) = x^3(x - 2y) - y^3(2x - y)\).

Key Concepts

Algebra

Algebra is a branch of mathematics that deals with symbols and rules for manipulating those symbols. These symbols often represent quantities without fixed values, known as variables. Understanding the basics of algebra is crucial because it helps in solving equations and understanding algebraic expressions. In this exercise, we're given a polynomial expression:
- A polynomial is a mathematical expression consisting of variables and coefficients, involving operations like addition, subtraction, multiplication, and non-negative integer exponents of variables.
- In the given problem, we have a fourth-degree polynomial, which means it includes a term raised to the fourth power.
When factoring polynomials in algebra, the goal is to express the polynomial as a product of simpler polynomials. This helps simplify expressions, solve equations, and understand relationships between variables.

Difference of Squares

The difference of squares is a special case in algebra where a polynomial takes the form of two squared terms subtracting from each other.
- The difference of squares formula is given by: \[ a^2 - b^2 = (a - b)(a + b) \] This formula makes it easy to factor expressions that fit this pattern.
- Our original expression brings this concept into play after the regrouping step. When we see terms like \((x^4 - y^4)\), it’s a hint that we can apply the difference of squares formula.
So remember, anytime you spot a squared term subtracted from another squared term, consider the difference of squares method. It’s a neat, efficient tool for simplifying algebraic expressions.

Factoring by Grouping

Factoring by grouping is an effective technique for simplifying polynomials, especially when they don’t fit other well-known formulas. It involves rearranging and grouping parts of the expression to find common factors.
- Begin by grouping terms that have common factors. In the exercise, we grouped as \[ (x^4 - 2x^3y) + (2xy^3 - y^4) \].
- This helps us notice commonalities, even when they aren’t immediately apparent. From these groups, factor out the greatest common factor in each.
Factoring by grouping can feel a bit like solving a puzzle. Each step helps reveal the structure hidden within a polynomial. With practice, you'll get better at spotting suitable groupings and using this method to simplify algebraic expressions efficiently.