Problem 51 Simplify. $$ -4 x 2-3 x y+7+... [FREE SOLUTION]

91影视

Elementary Algebra

John Redden

$Math Studyset 91影视 Explanations$ Math

2011 Edition

Chapter 2: Problem 51

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

Short Answer

Expert verified

The simplified expression is $-4x^2 - 8xy + 4xz + 4$.

Step by step solution

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 $xz$: $4xz$.

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:\[-4x^2 - 8xy + 4xz + 4\].

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:\[-4x^2 - 8xy + 4xz + 4\].

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

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

Short Answer

Step by step solution

Identify Like Terms

Combine Like Terms

Write the Simplified Expression

Key Concepts

Understanding Like Terms

Exploring Polynomial Expressions

Combining Constants in Expressions

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Applied Mathematics

Theoretical and Mathematical Physics

Decision Maths

Geometry

Mechanics Maths

Study anywhere. Anytime. Across all devices.