Problem 12 - Algebra 2

Question

Simplify. $$ (x+1)\left(x^{2}-2 x+3\right) $$

Step-by-Step Solution

Answer

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

Step 1: Distribute the First Term

First, distribute the term \((x)\) from the first binomial \((x+1)\) to each term in the trinomial \((x^2 - 2x + 3)\). This gives us:\[x imes (x^2 - 2x + 3) = x^3 - 2x^2 + 3x\]

Step 2: Distribute the Second Term

Next, distribute the term \((1)\) from the first binomial \((x+1)\) to each term in the trinomial \((x^2 - 2x + 3)\). This gives us:\[1 imes (x^2 - 2x + 3) = x^2 - 2x + 3\]

Step 3: Combine Like Terms

Combine the results from Step 1 and Step 2. Align the terms according to their degrees to simplify the expression:\[x^3 - 2x^2 + 3x + x^2 - 2x + 3\]Combine like terms:- Combine \(-2x^2\) and \(+x^2\) to get \(-x^2\).- Combine \(+3x\) and \(-2x\) to get \(+x\).The final simplified expression is:\[x^3 - x^2 + x + 3\]

Key Concepts

Distributive Property

The distributive property is a foundational concept in algebra. It allows you to multiply a single term by a group of terms inside parentheses. This property is expressed as \[ a(b + c) = ab + ac \] This means that you multiply the single term outside the parenthesis by each term inside the parenthesis separately. For example, in the expression \((x+1)(x^2-2x+3)\), the distributive property is applied twice:

• First, multiply \( x \) by each term in the polynomial \((x^2 - 2x + 3)\), which results in \( x^3 - 2x^2 + 3x \).
• Second, multiply \( 1 \) by each term in the polynomial \((x^2 - 2x + 3)\), resulting in \( x^2 - 2x + 3 \).

This step-by-step approach helps to ensure that all terms are correctly accounted for, leading to a consistent and simplified expression. Understanding the distributive property is crucial for simplifying more complex algebraic expressions.

Combining Like Terms

Combining like terms is an important step in simplifying polynomial expressions. After you have expanded an expression using the distributive property, you'll often end up with multiple terms that are similar. Like terms are terms that contain the same variable raised to the same power. For example:

• In \( x^3 - 2x^2 + 3x + x^2 - 2x + 3 \), the like terms are aligned based on their power.
• Combine \(-2x^2\) and \(+x^2\) to simplify to \(-x^2\).
• Combine \(+3x\) and \(-2x\) to simplify to \(+x\).

By combining like terms, you can reduce the expression to its simplest form, which is more concise and easier to interpret. In our example, the simplified form of the expression is \(x^3 - x^2 + x + 3\). Properly combining like terms helps in managing expressions for further operations and analysis.

Binomials and Trinomials

A binomial is a polynomial with two terms, and a trinomial is a polynomial with three terms. These structures are the basic building blocks in algebra, and understanding them is key to polynomial simplification. In the example \((x+1)(x^2-2x+3)\):

• The expression \((x+1)\) is a binomial, containing two terms: \(x\) and \(1\).
• The expression \((x^2 - 2x + 3)\) is a trinomial, containing three terms: \(x^2\), \(-2x\), and \(3\).

Working with binomials and trinomials involves multiplying them using the distributive property and simplifying the result by combining like terms. Recognizing and correctly applying operations to these polynomials are essential skills in algebra. Practicing these techniques makes it easier to handle more complex polynomial expressions effectively.