Problem 55 - Elementary Algebra

Question

Simplify. $$ x 2 x+1+6 x-242 x 2-7 x-4 $$

Step-by-Step Solution

Answer

The simplified expression is \(x^3 - x^2 - x - 4\).

Step 1: Identify Like Terms

The given expression is \(x^2(x + 1) + 6x - 2x^2 - 7x - 4\). Start by identifying and combining like terms. The expression involves polynomials' terms, so look for terms with the same variable and power.

Step 2: Distribute Terms

Distribute the \(x^2\) in the term \(x^2(x + 1)\) to simplify this part of the expression: \[x^2(x + 1) = x^3 + x^2\].Now our expression is: \[x^3 + x^2 + 6x - 2x^2 - 7x - 4\].

Step 3: Combine Like Terms

Look for and combine like terms in the expression:1. Combine \(x^2\) terms: \(x^2 - 2x^2 = -x^2\).2. Combine \(x\) terms: \(6x - 7x = -x\).After combining, the simplified expression is:\[x^3 - x^2 - x - 4\].

Step 4: Write Final Simplified Expression

The fully simplified expression is: \[x^3 - x^2 - x - 4\].This expression has been simplified to its minimal terms with all like terms combined.

Key Concepts

Combining Like Terms

When simplifying algebraic expressions, combining like terms is a crucial step. Like terms are terms that have the same variable raised to the same power. This means you can only combine terms with identical variable parts, such as \(x^2\) with \(x^2\) or \(x\) with \(x\). In our expression, combining these terms helps streamline the equation and simplify further operations.

Consider the process:

• Identify terms that have the same variable and power, such as \(x^2\) and \(-2x^2\).
• Add or subtract the coefficients of these terms to simplify. For example, \(x^2 - 2x^2\) results in \(-x^2\).

This step reduces the complexity of your algebraic expression, making it easier to solve or further manipulate.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and mathematical operations. They can include terms with variables raised to different powers, constants, and coefficients. Understanding algebraic expressions helps in manipulating and simplifying equations in mathematics.

In our problem, the expression is a mix of terms like \(x^2(x + 1)\), \(6x\), \(-2x^2\), \(-7x\), and \(-4\). Here's what you need to know:

• Variables: Symbols like \(x\) that represent unknown values.
• Coefficients: Numbers that multiply the variables, such as 6 in \(6x\).
• Constants: Numbers without variables, like \(-4\) in our expression.

The goal is to simplify algebraic expressions by applying operations consistently and combining like terms whenever possible.

Polynomials

Polynomials are a specific type of algebraic expression that consist of sums of terms, each including a variable raised to a whole number power. They are fundamental in algebra for solving equations and modeling relationships. In this problem, we're dealing with polynomials as we simplify terms with \(x^3\), \(x^2\), and \(x\).

Here's a quick breakdown:

• Degree of a Polynomial: The highest power of the variable in the polynomial. Our final simplified polynomial is \(x^3 - x^2 - x - 4\), with a degree of 3.
• Monomials, Binomials, and Trinomials: Based on the number of terms, a polynomial can be classified, e.g., monomials (one term), binomials (two terms), and so on.

Understanding polynomials helps in recognizing patterns and simplifying complex expressions by organizing terms.

Distribution in Algebra

The distributive property is an essential tool in algebra for breaking down expressions into simpler parts. When you distribute, you multiply a single term across terms inside a parenthesis. In the exercise, we applied distribution to simplify \(x^2(x + 1)\).

Here's how it works:

• Multiply \(x^2\) by each term inside the parentheses. So, \(x^2 \times x = x^3\) and \(x^2 \times 1 = x^2\).
• Combine these results to get \(x^3 + x^2\).

This method helps unpack expressions making easier to identify and combine like terms later. Distribution rearranges and simplifies how we look at complex expressions, setting the stage for further simplification steps.