Problem 72 - Algebra and Trigonometry

Question

31–76 ? Factor the expression completely. $$ (x+1)^{3} x-2(x+1)^{2} x^{2}+x^{3}(x+1) $$

Step-by-Step Solution

Answer

The completely factored expression is \(x(x+1)\).

Step 1: Analyze the expression

The given expression is \[(x+1)^{3} x - 2(x+1)^{2} x^{2} + x^{3}(x+1)\]Observe that each term contains both \(x\) and \((x+1)\) as common factors.

Step 2: Factor out the common terms

Factor the common terms \(x(x+1)\) from the entire expression:\[x(x+1)ig((x+1)^2 - 2x(x+1) + x^2ig)\]

Step 3: Expand and simplify the inner expression

Now focus on the inner expression:\((x+1)^2 - 2x(x+1) + x^2\).Expand each term:\((x+1)^2 = x^2 + 2x + 1\)\(-2x(x+1) = -2x^2 - 2x\)Substitute and simplify:\[x^2 + 2x + 1 - 2x^2 - 2x + x^2 = 1\]

Step 4: Rewrite the factored expression

Since the inner expression simplifies to 1, rewrite the factored expression as:\[x(x+1) \times 1 = x(x+1)\]

Step 5: Verify the simplification

Verify by expanding \(x(x+1)\) back to the original form to ensure correctness:\[x(x+1) = x^2 + x\] and note that the simplification step was correct based on the original expression.

Key Concepts

Polynomial Expressions

Polynomial expressions are algebraic expressions made up of variables and coefficients, constructed using only the operations of addition, subtraction, multiplication, and non-negative integer exponents. In the given exercise, the polynomial is:

• \((x+1)^3 x - 2(x+1)^2 x^2 + x^3(x+1)\)

Here, each term in the polynomial is expressed in terms of the variable \(x\) and combines it with another binomial \((x+1)\). Understanding how to manipulate these expressions is fundamental to performing operations like factoring, which is our main focus in this scenario. By dissecting polynomial expressions into terms, we can identify common elements and patterns critical for further algebraic operations. Remember, practice with polynomials helps you get familiar with different forms and structures you might encounter.

Common Factors

A common factor is an element that appears in each term of an expression or equation. In factorizing polynomial expressions, identifying common factors is a crucial step because it helps simplify the expression into a more manageable form.

• In the original expression \((x+1)^3 x - 2(x+1)^2 x^2 + x^3(x+1)\), each term includes both \(x\) and \((x+1)\) as factors.
• By factoring out these common elements, we reduce the complexity of the expression.

This means rewriting the polynomial as:\[x(x+1)\big((x+1)^2 - 2x(x+1) + x^2\big)\]Recognizing common factors can be a powerful tool in simplifying and understanding the structure of polynomial expressions. It allows us to make expressions simpler and clearer by pulling out shared components, which eases further simplifications.

Simplification

Simplification involves reducing an expression to its simplest form. It makes expressions easier to handle and more concise. In our example, after factoring out the common elements \(x(x+1)\), we are left with the inner expression:

• \((x+1)^2 - 2x(x+1) + x^2\)

The process of simplifying involves expanding terms, combining like terms, and performing arithmetic operations to condense the expression:

• Expand \((x+1)^2\) to get \(x^2 + 2x + 1\).
• Handle \(-2x(x+1)\) to yield \(-2x^2 - 2x\).
• Add the expanded terms: \(x^2 + 2x + 1 - 2x^2 - 2x\).

This results in simplifying the inner expression to just \(1\), leaving us with a very tidy expression, \(x(x+1)\). Simplification not only shows the elegance of algebra but also prepares the expression for further algebraic manipulation if needed.

Algebraic Manipulation

Algebraic manipulation is the process of using algebraic techniques to rearrange and solve expressions or equations. It includes factoring, expanding, substituting, and simplifying, allowing us to work through complex problems more efficiently. In this exercise, we started with a cumbersome polynomial expression, breaking it into smaller parts to make it easier to handle. Some algebraic manipulation techniques used here include:

• Factoring common terms, which facilitated the process by lumping similar elements together.
• Expanding and combining like terms within the inner expression to distill it down to the simplest form.

Each step involved deliberate algebraic manipulation to ensure that we arrived at an accurate and simplified final form. Knowing these strategies is essential for effectively working with algebraic expressions, making you more adept at solving a variety of algebraic problems.