Problem 80 - Elementary Algebra

Question

Simplify the algebraic expressions for the following problems. $$ x^{2}+3 x-4-4 x^{2}-5 x-9+2 x^{2}-6 $$

Step-by-Step Solution

Answer

Question: Simplify the following algebraic expression: $$x^{2}+3 x-4-4 x^{2}-5 x-9+2 x^{2}-6$$ Answer: $$-x^{2}-2x-19$$

Step 1: Identify like terms

First, we need to identify the terms with the same degree in the expression, such as the squared terms, linear terms, and constants. Expression: $$ x^{2}+3 x-4-4 x^{2}-5 x-9+2 x^{2}-6 $$ Squared terms: \(x^{2}\), \(-4x^{2}\), \(2x^{2}\) Linear terms: \(3x\), \(-5x\) Constants: \(-4\), \(-9\), \(-6\)

Step 2: Combine like terms

Next, we need to combine the like terms by adding or subtracting them based on their coefficients. For squared terms: $$ x^{2}+(-4x^{2})+(2x^{2})=(1-4+2)x^{2}=-x^{2} $$ For linear terms: $$ 3x+(-5x)= (3-5)x=-2x $$ For constants: $$ -4+(-9)+(-6)= -4-9-6= -19 $$

Step 3: Write the simplified expression

Lastly, we write the simplified expression by putting together the combined terms from the previous step. Simplified Expression: $$ -x^{2}-2x-19 $$ Therefore, the simplified form of the given algebraic expression is: $$ -x^{2}-2x-19 $$

Key Concepts

Simplifying Expressions

When simplifying algebraic expressions, the main goal is to make them as straightforward as possible. Think of simplifying as tidying up mathematical "clutter." By reorganizing the terms, you can make complex expressions more manageable.

To simplify an expression, you must first look for like terms. Like terms are terms that can be combined because they have the same variable raised to the same power. For instance, in the expression \( x^2 + 3x - 4 - 4x^2 - 5x - 9 + 2x^2 - 6 \), you can group the terms that have \( x^2 \) or \( x \) separately. Constants, like numbers without variables, are also grouped together to tidy up the expression. By combining these, the expression becomes much easier to deal with.

Like Terms

Like terms are essential components in algebra because they help you combine and simplify expressions. A like term means having the same variable raised to the same exponent. Let's break it down:

• Consider terms such as \( x^2, -4x^2, \) and \( 2x^2 \).
• These are like terms because they each contain the variable \( x \) raised to the power of 2.
• Other examples include linear terms like \( 3x \) and \( -5x \), which share the same variable and exponent (1 in this case).
• Constants, such as \(-4, -9, \) and \(-6\), are also considered like terms because they are numbers without any variables.

By understanding and identifying like terms, you can combine them by adding or subtracting their coefficients to simplify your algebraic expressions efficiently.

Polynomial Simplification

Polynomial simplification involves reducing a polynomial to its simplest form by collecting like terms and performing arithmetic operations. Polynomials are expressions that include variables, constants, and coefficients. Simplifying involves a few straightforward steps:

• Identify and group the like terms, which can make lengthy expressions easier to manage.
• Combine the coefficients of like terms.
• Make sure to carry out operations like addition or subtraction carefully to avoid any mistakes.

For example, let's take the given expression \( x^2 + 3x - 4 - 4x^2 - 5x - 9 + 2x^2 - 6 \). First, group and simplify the squared terms:

• Combine \( x^2, -4x^2, \) and \( 2x^2 \) to get \(-x^2\).

Then, focus on the linear terms and constants:

• Combine \( 3x \) and \(-5x\) to get \(-2x\).
• Sum up the constants: \(-4, -9, \) and \(-6\) to simplify to \(-19\).

By performing these steps, you condense the polynomial into its simplest form: \( -x^2 - 2x - 19 \). This simplification provides clarity and makes further mathematical work more manageable.