Problem 82 - Elementary Algebra

Question

Simplify the algebraic expressions for the following problems. $$ 2 x^{2}-x-\left(3 x^{2}-4 x-5\right) $$

Step-by-Step Solution

Answer

Answer: The simplified form of the given algebraic expression is $$-x^2 + 3x + 5$$.

Step 1: Remove Parentheses by Distributing the Negative Sign

Start by distributing the negative sign outside the parentheses to simplify the expression. This will eliminate the parentheses and make it easier to combine like terms. $$ 2x^2 - x - (3x^2 - 4x - 5) = 2x^2 - x - 3x^2 + 4x + 5 $$

Step 2: Combine Like Terms

Now that the parentheses have been removed, identify and combine the terms that have the same variable with the same exponent. $$ (2x^2 - 3x^2) + (-x + 4x) + 5 = -x^2 + 3x + 5 $$ The simplified algebraic expression is: $$ -x^2 + 3x + 5 $$

Key Concepts

The Distributive Property

When working with algebraic expressions like \(2x^2 - x - (3x^2 - 4x - 5)\), the distributive property is a crucial step in simplifying.The main purpose of the distributive property here is to eliminate parentheses by distributing any signs or numerals outside the parentheses across each term within it.

• In our example, the expression inside the parentheses is being subtracted: \(-(3x^2 - 4x - 5)\).
• To distribute the negative sign, you change the signs of each term within the parentheses: \(-3x^2, +4x, +5\).

This transforms the equation into a much more manageable form. By using this technique, you prepare the expression for the next stage: combining like terms.

Combining Like Terms

The next key step in simplifying algebraic expressions is combining like terms. After the distributive property has been applied, you group together terms that have the same variable raised to the same power.Let's look at the expression after distributing: \(2x^2 - x - 3x^2 + 4x + 5\).

• Identify like terms: \(2x^2\) and \(-3x^2\); \(-x\) and \(+4x\).
• Combine the like terms. For the \(x^2\) terms: \(2x^2 - 3x^2 = -x^2\).
• For the \(x\) terms: \(-x + 4x = 3x\).
• Constant terms \(+5\) remains unchanged and standalone.

This results in the simplified expression \(-x^2 + 3x + 5\). Combining like terms simplifies expressions efficiently, allowing for clearer mathematical operations or solving equations.

Introduction to Polynomials

Polynomials are algebraic expressions that consist of variables and coefficients, combined using addition, subtraction, and multiplication. They do not include division by a variable.In the expression given \(-x^2 + 3x + 5\):

• Each part separated by a plus or minus sign is a term, so there are three terms: \(-x^2\), \(+3x\), and \(+5\).
• Terms like \(x^2\) are called monomials because they include only one variable along with their coefficient.
• The degree of a polynomial is the highest exponent of any term. Here, \(-x^2\) is the highest degree term (degree 2), so this is a second-degree polynomial.

Understanding polynomials helps in recognizing patterns and performing operations such as addition or multiplication. Knowing how to rearrange and simplify them is essential for advanced algebra.