Problem 71 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The expression factors to \(3(x-1)(x+2)\).

Step 1: Identify a Common Factor

Look at both terms in the expression \((x-1)(x+2)^2 - (x-1)^2(x+2)\). Notice that both terms have \((x-1)\) and \((x+2)\) as common factors. Therefore, we can factor out the greatest common factor, which is \((x-1)(x+2)\).

Step 2: Factor Out the Common Factor

Factor \((x-1)(x+2)\) from the whole expression:\[(x-1)(x+2)^2 - (x-1)^2(x+2) = (x-1)(x+2) \left[ (x+2) - (x-1) \right].\]

Step 3: Simplify the Factored Expression

Simplify the expression inside the brackets. Calculate \((x+2) - (x-1)\):\[ (x+2) - (x-1) = x + 2 - x + 1 = 3. \]

Step 4: Write the Final Factored Expression

Combine all parts into the final expression:\[(x-1)(x+2)(3).\] This simplifies to \[3(x-1)(x+2).\]

Key Concepts

Common Factors

A common factor between terms in an algebraic expression is a quantity that divides each term in the expression without leaving a remainder. When two terms share common factors, these can be factored out to simplify the expression. In the case of the expression

• \((x-1)(x+2)^2 - (x-1)^2(x+2)\)

both terms include

• \( (x-1) \)
• \( (x+2) \)

as common factors. This means that the greatest common factor here is

• \((x-1)(x+2)\)

which can be factored out from the entire expression. Factoring out common factors is a powerful technique that simplifies polynomials, reducing them into products that can be more easily managed or solved in further steps. By identifying the common factors correctly, you make the solving process significantly easier.

Polynomial Expressions

Polynomial expressions are sums of terms that are made up of constants, variables, and exponents. An expression like

• \((x-1)(x+2)^2 - (x-1)^2(x+2)\)

can be considered a polynomial expression. Understanding the structure of polynomial expressions is key to simplifying or factoring them. Polynomials often include similar terms that can be grouped together or reduced using mathematical operations. Within their complex forms, looking for common variables or combinations of variables will help identify how to reduce or factor them effectively. Simplifying a polynomial might involve:

• Factoring out common factors.
• Applying operations to reduce similar terms.
• Reconstructing the expression to reveal simpler forms.

These steps make it possible to analyze and solve polynomial problems with greater ease.

Factoring Techniques

Factoring techniques involve rewriting a polynomial as a product of simpler polynomials. Each technique is effective under certain circumstances, and understanding when to use which technique is crucial. The expression

• \((x-1)(x+2)^2 - (x-1)^2(x+2)\)

is factored by recognizing common factors, a basic yet fundamental technique for simplifying expressions. The steps are as follows:

• Identify the greatest common factors: \((x-1)(x+2)\)
• Factor these out, reducing complexity.
• Simplify the resultant expression.

Partial and complete factoring are essential in solving polynomial equations, turning complex expressions into products of simpler units. This aids in understanding the roots or solutions of polynomial equations. By mastering different factoring techniques such as grouping, using special identities, and finding roots, you solidify your problem-solving skills in algebra.