Problem 78 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The expression factors to \((x+1)^2 x(x^2-x+1)\).

Step 1: Identify Common Factors

First, observe the expression \((x+1)^{3} x - 2(x+1)^{2} x^{2} + x^{3}(x+1)\). Each term includes \((x+1)\) and a power of \(x\). Let's express it as follows:- \((x+1)^3 x\) is \((x+1)^3 \cdot x\),- \(-2(x+1)^2 x^2\) is \(-2(x+1)^2 \cdot x^2\),- \(x^3(x+1)\) is \(x^3 \cdot (x+1)\).Identify that \((x+1)\) is a common factor in every term, and the smallest exponent among these is \((x+1)^2\). The smallest power of \(x\) present is \(x^1\). Therefore, the common factor to factor out is \((x+1)^2 \cdot x\).

Step 2: Factor Out the Common Factor

Factor \((x+1)^2 \cdot x\) out of the expression.\[((x+1)^2 \cdot x)\left[(x+1) - 2x + x^2\right]\].Here, we factor as:- From \((x+1)^3 \cdot x\), removing \((x+1)^2 \cdot x\) leaves \((x+1)\).- From \(-2(x+1)^2 \cdot x^2\), removing \((x+1)^2 \cdot x\) gives \(-2x\).- From \(x^3 \/ (x+1)\), removing \((x+1)^2 \cdot x\) results in \(x^2\).

Step 3: Simplify the Remaining Expression

Simplify the expression inside the parentheses:\[(x+1) - 2x + x^2\]which simplifies to:\[x^2 - x + 1\].Thus, the factored expression is \((x+1)^2 \cdot x \cdot (x^2 - x + 1)\).

Key Concepts

Common Factors

When factoring polynomials, one of the primary steps is to identify any common factors present among the terms. A **common factor** is an expression that can be divided out of each term without leaving a remainder.

In the given exercise, each term contains the binomial (\((x+1)\)) and a power of \(x\). Our task is to find the largest common factor that is present in all terms:

• \((x+1)^3 x\) has the factor (\((x+1)^3 \cdot x\)).
• -2(\(x+1)^2 x^2\) includes (\((x+1)^2 \cdot x^2\)).
• \(x^3(x+1)\) can be expressed as (\(x^3 \cdot (x+1)\)).

From these, we can observe that the smallest power of (\((x+1)\)) present in all terms is (\((x+1)^2\)). Similarly, the smallest power of \(x\) present is (\(x^1\)). Hence, the common factor we factor out is (\((x+1)^2 \cdot x\)).

This step simplifies the polynomial and sets the stage for finding simpler expressions.

Factoring Techniques

Once the common factor has been identified, the next step is to utilize various **factoring techniques** to simplify the expression further. Removing the common factor significantly reduces the complexity of the polynomial.

Let's retrace the steps of factoring out our identified common factor from each term:

• From (\((x+1)^3 x\)), taking out (\((x+1)^2 \cdot x\)) leaves (\((x+1)\)) alone.
• From (-2(\(x+1)^2 x^2\)), removing (\((x+1)^2 \cdot x\)) results in (-2x).
• Finally, from (\(x^3(x+1)\)), the common factor removal yields (\(x^2\)).

After factoring out the common factor, we apply other techniques like looking for patterns or simply the direct simplification of remaining expressions inside the brackets. This technique helps to rewrite the polynomial in a simpler and more manageable form, as seen in our exercise.

Simplifying Expressions

**Simplifying expressions** is the final touch to make the polynomial as simple as possible. After factoring out, we are often left with an expression inside parentheses that can be easily simplified.

In our case, the expression within parentheses: (\[(x+1) - 2x + x^2\]) needs further simplification.

• We need to combine like terms in a systematic manner.
• This results in rewriting as (\[x^2 - x + 1\]).

Once this expression inside the brackets is simplified, we can state the final factored expression as (\[(x+1)^2 \cdot x \cdot (x^2 - x + 1)\]).

These simplification steps ensure the polynomial is expressed in its most reduced form, making it easier to interpret, work with in further algebraic operations, or use practically in problems needing this polynomial's factorized form.