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Chapter 0: Problem 5

\((x+1)(x-3)\)

Short Answer

Expert verified
The simplified and expanded form of the given expression \((x+1)(x-3)\) is \(x^2 - 2x - 3\).

Step by step solution

01

Apply the Distributive Property

To expand the expression, we will multiply each term in the first parentheses with each term in the second parentheses, following the distributive property formula: \((a+b)(c+d) = ac + ad + bc + bd\). Doing this for our expression will give: \((x+1)(x-3) = x\cdot x + x\cdot (-3) + 1\cdot x + 1\cdot (-3)\)
02

Simplify the Multiplication Results

Next, let's simplify the four multiplication results we obtained in step 1: \(x\cdot x = x^2\) \(x\cdot (-3) = -3x\) \(1\cdot x = x\) \(1\cdot (-3) = -3\) Now, rewrite the expanded expression with these simplified terms: \(x^2 - 3x + x - 3\)
03

Combine Like Terms

Now, let's combine the like terms (i.e., terms with the same variable and exponent). The expression currently stands as \(x^2 - 3x + x - 3\). Combine the two like terms, which are the '-3x' and 'x' terms: \(-3x + x = -2x\) Finally, write down the simplified and expanded form of the expression: \(x^2 - 2x - 3\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Distributive Property
The distributive property is a fundamental concept in algebra that allows us to simplify and expand expressions. It states that \[(a + b)(c + d) = ac + ad + bc + bd\]This property is especially useful when dealing with polynomials and other expressions involving parentheses. In our exercise, \((x+1)(x-3)\), we apply this property by multiplying each term in the first parenthesis with each term in the second. This involves:
  • Multiplying \(x\) by \(x\) and \(-3\)
  • Multiplying \(1\) by \(x\) and \(-3\)
Following these steps helps unravel the expression from its factored form. Utilizing the distributive property correctly is essential in moving towards a simplified version of any given polynomial.
Simplifying Expressions
Simplifying expressions helps make complex equations easier to handle. It involves performing basic arithmetic or algebraic manipulations. In our case, after applying the distributive property, we have:
  • \(x \cdot x = x^2\)
  • \(x \cdot (-3) = -3x\)
  • \(1 \cdot x = x\)
  • \(1 \cdot (-3) = -3\)
Each of these calculations breaks down the original expression into individual terms: \[x^2 - 3x + x - 3\]This simplification allows us to clearly see all parts of the expression, paving the way for easier manipulation in later steps. Simplifying before combining terms ensures we don鈥檛 overlook any needed operations or errors.
Combining Like Terms
Combining like terms is the final step in simplifying polynomial expressions. This involves grouping and consolidating terms that have identical variable parts. In our example, we have the expanded form:\[x^2 - 3x + x - 3\]In this expression, \(-3x\) and \(x\) are like terms because they both have the variable \(x\) to the same power. Adding these together results in:
  • \(-3x + x = -2x\)
Now, the expression is simplified to \[x^2 - 2x - 3\].By combining like terms, we reduce clutter and clarify the structure of the polynomial, making it easier to interpret and use in further calculations.

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