Problem 40 - College Algebra

Question

For the following exercises, find the sum or difference. $$(x-1)\left(x^{2}-2 x+1\right)$$

Step-by-Step Solution

Answer

The simplified expression is \(x^3 - 3x^2 + 3x - 1\).

Step 1: Distribute the Terms

To solve the given expression, distribute the term \((x-1)\) across the polynomial \((x^2 - 2x + 1)\). This means multiplying \(x\) by each term in the polynomial and then \(-1\) by each term in the polynomial.

Step 2: Multiply x with Polynomial Terms

Multiply \(x\) with each term in \(x^2 - 2x + 1\):\[ x \cdot (x^2) = x^3 \] \[ x \cdot (-2x) = -2x^2 \] \[ x \cdot 1 = x \] The result is \(x^3 - 2x^2 + x\).

Step 3: Multiply -1 with Polynomial Terms

Multiply \(-1\) with each term in \(x^2 - 2x + 1\):\[ -1 \cdot (x^2) = -x^2 \] \[ -1 \cdot (-2x) = 2x \] \[ -1 \cdot 1 = -1 \] The result is \(-x^2 + 2x - 1\).

Step 4: Combine Like Terms

Now, add the results from Step 2 and Step 3 together. Combine like terms: \[ x^3 - 2x^2 + x - x^2 + 2x - 1 \] Simplify by combining like terms: \[ x^3 - 3x^2 + 3x - 1 \]. This is the simplified form of the expression.

Key Concepts

Distributive Property

The distributive property is a fundamental method in algebra used to simplify expressions and solve equations. When you encounter an expression like \((x-1)(x^2 - 2x + 1)\), you need to multiply each term in the parenthesis \( (x-1) \) by the terms in the polynomial \( (x^2 - 2x + 1) \). This is the essence of the distributive property.Here’s what you should do:

• Multiply the first term in the first parenthesis by each term in the second parenthesis.
• Repeat this process with the next term in the first parenthesis.

For our example:

• Multiply \( x \) by \(x^2\), resulting in \(x^3\).
• Multiply \( x \) by \(-2x\), which gives \(-2x^2\).
• Multiply \( x \) by \(1\), resulting in \(x\).
• Then move to \(-1\), and multiply it by each term in \(x^2 - 2x + 1\).
• This gives us \(-x^2\), \(2x\), and \(-1\).

This step-by-step approach helps you to systematically apply the distributive property without missing any terms. By following these steps, you will manage to simplify algebraic expressions efficiently.

Combining Like Terms

Once you've distributed all the terms, the next process is simplifying the expression by combining like terms. This is a critical step in making algebraic expressions easier to understand. But what are like terms?Like terms are terms in an expression that have the exact same variables raised to the same powers. In the expression \[ x^3 - 2x^2 + x - x^2 + 2x - 1 \], we can identify our like terms.Follow these steps to combine them:

• Identify terms with the same variable and power. Here, you have \(-2x^2 \) and \(-x^2\) as like terms, so combine them to get \(-3x^2\).
• Similarly, combine \(x\) and \(2x\) to result in \(3x\).
• Since \(x^3\) and \(-1\) have no like terms, they remain unchanged.

After combining like terms, the simplified expression becomes \[ x^3 - 3x^2 + 3x - 1 \].This process not only aids in simplifying the expression but also makes it easier to work with in further mathematical operations like solving equations.

Algebraic Expressions

Algebraic expressions are a combination of variables, numbers, and operations (like addition, subtraction, multiplication, etc.). They are a core component of algebra, helping us represent real-world problems in mathematical form.In the expression \((x-1)(x^2 - 2x + 1)\), you can identify:

• Variables: \( x \) is the variable, representing unknown values.
• Numbers: the integers \(-1\), \(2\), and \(1\) are constants that modify the variables.
• Operators: including multiplication (used in the distributive process) and addition/subtraction (used when combining terms).

Understanding algebraic expressions is crucial because it allows you to negotiate word problems and perform real-life calculations. By mastering operations like distribution and combining like terms, students can break down complex problems into more manageable, simpler parts. Developing these foundational skills is essential for success in higher-level math courses and various applied fields.