Problem 51 - Elementary Algebra

Question

Simplify. $$ -4 x 2-3 x y+7+4 x z-5 x y-3 $$

Step-by-Step Solution

Answer

The simplified expression is \(-4x^2 - 8xy + 4xz + 4\).

Step 1: Identify Like Terms

Write down all terms and identify which ones can be grouped together. The expression is: \[-4x^2 - 3xy + 7 + 4xz - 5xy - 3\].- Constants: \(7\) and \(-3\).- Terms with \(x^2\): \(-4x^2\).- Terms with \(xy\): \(-3xy\) and \(-5xy\).- Terms with \(xz\): \(4xz\).

Step 2: Combine Like Terms

Combine the constants and like terms as follows:- Combine the constants: \(7 - 3 = 4\).- Combine the \(xy\) terms: \(-3xy - 5xy = -8xy\). The new expression, after combination, becomes:\[-4x^2 - 8xy + 4xz + 4\].

Step 3: Write the Simplified Expression

List the terms of the expression in standard polynomial form. Since the terms are already organized by degrees and types, your expression is:\[-4x^2 - 8xy + 4xz + 4\].

Key Concepts

Understanding Like Terms

In algebra, like terms are terms that have the same variables raised to the same power. Identifying like terms is essential for simplifying algebraic expressions.

Here's what matters when considering like terms:

• Each term must have identical variable parts. For example, in the expression \(-3xy - 5xy\), both terms have the exact variables \(xy\). Hence, they are like terms.
• Only the coefficients, which are the numbers in front of the variables, can differ. For instance, -3 and -5 in the example are coefficients.
• Like terms can be added or subtracted directly. Thus, \(-3xy - 5xy = -8xy\) because we combine the coefficients.

Recognizing like terms enables us to simplify expressions effectively, paving the way for understanding more complex algebraic concepts.

Exploring Polynomial Expressions

A polynomial expression consists of variables, coefficients, and exponents, combined using addition and subtraction.

Key characteristics of polynomial expressions include:

• Each part of a polynomial separated by a plus or minus sign is termed as a 'term.' For example, \(-4x^2\) and \(4xz\) are individual terms in our expression.
• The degree of a typical polynomial term is determined by the sum of the exponents of its variables. For example, in \(-4x^2\), the degree is 2, while in \(-3xy\), the degree is 2 since \(x^1y^1\) gives \(1+1=2\).
• Polynomials are organized in standard form starting with the highest degree on the left and moving to the lowest degree on the right.

Grasping the structure of polynomial expressions simplifies not just solving but also visualizing algebraic problems efficiently.

Combining Constants in Expressions

Combining constants is an easy yet crucial step when simplifying expressions.

Constants are the numbers in expressions that do not have any variables attached, such as 7 and -3 in our expression. Here's how to combine them:

• Add or subtract constants directly, just like when you deal with simple numbers. In our case, \(7 - 3 = 4\).
• This combined constant becomes a standalone term in the simplified expression.

Understanding how to handle constants correctly ensures the overall simplification is accurate, contributing to clearer and more manageable algebraic expressions.