Problem 32 - Elementary Algebra

Question

Simplify. $$ 2 x-3 x+5-x x-3 $$

Step-by-Step Solution

Answer

\(-x^2 - x + 2\)

Step 1: Simplify Like Terms

Begin by simplifying like terms. The expression is \(2x - 3x + 5 - xx - 3\). Like terms include \(2x\) and \(-3x\), which can be combined to give: \((2x - 3x) = -x\). Thus, the expression becomes \(-x + 5 - xx - 3\).

Step 2: Combine Constants

Next, combine the constant terms \(5\) and \(-3\) to simplify further. \(5 - 3 = 2\). This gives the intermediate expression: \(-x + 2 - xx\).

Step 3: Simplify Further by Subtracting \(xx\)

Now consider the term \(-xx\), which is a product of \(-1\) and \(x\cdot x\), simplifying to \(-x^2\). The expression now becomes \(-x - x^2 + 2\).

Step 4: Write in Standard Form

The standard form for polynomials is to list terms in order of decreasing degree. Reorder \(-x^2 - x + 2\) as this is the simplest form.

Key Concepts

Polynomial Expression

A polynomial expression is a sum or difference of terms that include variables raised to whole number exponents and constants. Understanding polynomials is key to mastering algebraic manipulation because they often appear in equations and expressions.
Polynomials have terms that are made up of:

• Coefficients: The numerical part of terms, such as the "2" in "2x".
• Variables: The unknowns represented by letters, such as "x".
• Exponents: The power to which a variable is raised, such as "x" raised to the 2 in "x^2".

In the given expression:
- Each part like "2x", "xx", and constants like "5" are considered terms of the polynomial. The operation to simplify involves combining these terms properly while adhering to polynomial rules, like ensuring terms involve only whole number exponents.

Like Terms

Like terms in algebra are terms that have the same variable raised to the same power. This is fundamental in simplifying expressions because it helps reduce the complexity of the expression.
For instance, in the expression

• "2x" and "-3x" are like terms because both have the same variable "x" raised to the first power.
• They can be combined into one term by simply adding or subtracting their coefficients — here, yielding "-x" because (2-3)x = -1x.

Terms like "xx" or "x^2" are not like terms with "x". They have different exponents and thus must be handled separately.

Combining Constants

Combining constants is the process of simplifying an expression by adding or subtracting the numbers that are not attached to any variables. Constants are plain numbers within a polynomial expression that do not multiply variables.
In the example expression:

• The constants "5" and "-3" can be directly combined because they are plain numbers without variables.
• Perform the arithmetic operation, which results in "2" (since 5 - 3 = 2).

This step is crucial because it helps in reducing the expression to a form that is much cleaner and easier to understand.

Standard Form in Algebra

Standard form in algebra refers to a way of writing expressions, especially polynomials, so they are presented in a conventional manner. When an expression is in standard form:

• Terms are typically ordered by the degree of the variable — from highest to lowest. The degree is essentially the highest power of the variable.
• For example, the expression "-x^2 - x + 2" is in standard form because terms are sorted from the highest power ("-x^2", power of 2) down to the constant term ("2", power of 0).

Writing in standard form ensures readability and uniformity, which is useful for further algebraic operations, making it easier to compare and manipulate expressions.