Problem 38 - Elementary Algebra

Question

Simplify. (Assume all denominators are nonzero.) $$ 5 x-5+20-9 x 2 x 2-15 x+25 $$

Step-by-Step Solution

Answer

The simplified expression is \(2x^2 - 19x + 40\).

Step 1: Identify Like Terms

First, identify and group the like terms in the expression: \(5x - 5 + 20 - 9x + 2x^2 - 15x + 25\). Recognize the terms involving \(x^2\), \(x\), and constant terms separately.

Step 2: Combine Like Terms

Combine like terms:- The \(x^2\) term is \(2x^2\) (since there are no other \(x^2\) terms).- Combine the \(x\) terms: \(5x - 9x - 15x = -19x\).- Combine the constant terms: \(-5 + 20 + 25 = 40\).

Step 3: Write the Simplified Expression

Substituting the results from Step 2, our simplified expression becomes:\[2x^2 - 19x + 40\].

Key Concepts

Understanding Like Terms

In algebra, a crucial concept is the understanding of 'like terms'. Like terms are terms that have the same variable raised to the same power. They are essential when it comes to simplifying expressions because they can be combined to form a single term. For example, in the expression \(5x - 9x + 2x^2\), the terms \(5x\) and \(-9x\) are like terms because they both involve the variable \(x\) to the first power. The term \(2x^2\), however, is not a like term with \(5x\) or \(-9x\) since it involves \(x\) squared. To identify like terms, look for these similar characteristics:

• Same variable name.
• Same exponent associated with the variable.

Once you have grasped this, simplification becomes a matter of basic addition and subtraction.

Combining Terms in Algebra

Combining terms involves simplifying an expression by adding or subtracting like terms. This process turns a long, complex expression into a more manageable one, making it easier to interpret and solve equations. When we refer to combining terms, we specifically mean:

• Collecting like terms and performing the arithmetic operations indicated on those terms.
• Ensuring that coefficients are added or subtracted correctly while keeping the variable part unchanged.

Let's consider the example from the original exercise: combining the \(x\) terms in the expression \(5x - 9x - 15x\):
First, identify them as like terms (all have the variable \(x\)). Then, add up the coefficients: \(5 - 9 - 15 = -19\). Therefore, the combined term becomes \(-19x\). This simplification helps us reduce excess clutter and arrive at a straightforward expression.

Exploration of Polynomials

Polynomials are expressions made up of multiple terms that are added or subtracted. These terms are usually made of variables raised to various powers, coefficients, and constant terms. An expression like \(2x^2 - 19x + 40\) is a polynomial:

• The first term, \(2x^2\), represents the quadratic term because it involves \(x\) squared.
• The second term, \(-19x\), is the linear term, which involves \(x\) to the first power.
• The last term, \(40\), is a constant term, with no variables present.

Understanding polynomials helps in recognizing how to manipulate these expressions with ease. The structure of polynomials allows us to classify them based on their degree (the highest exponent) and the number of terms they contain. This understanding proves valuable in tasks like factoring, finding roots, and graphing. Additionally, knowing how to manipulate polynomials is key in solving more complex algebraic equations.