Problem 32 Identify the terms of each polyn... [FREE SOLUTION]

Chapter 15: Problem 32

Identify the terms of each polynomial. $$ -a^{2} b+4 a^{2}-8 b+17 $$

Short Answer

Expert verified

The terms are oindent \text{\footnotesize $ -a^{2}b, \text{\footnotesize $ 4a^{2, \text{\footnotesize $ -8b, and \text{\footnotesize $ 17}}

Step by step solution

Understand the Polynomial Structure

A polynomial is an expression consisting of variables and coefficients combined using addition, subtraction, and multiplication. Each combination of variables and coefficients in a polynomial is called a term.

Identify the Terms by Breaking the Polynomial

To find the terms of the polynomial, separate each part of the polynomial that is added or subtracted. In this case, the polynomial is oindent \text{\footnotesize $ -a^{2}b + 4a^{2} - 8b + 17$}

List All Identified Terms

After breaking down the polynomial, the individual terms are:oindent \text{\footnotesize $ -a^{2}b, \text{\footnotesize $ 4a^{2, \text{\footnotesize $ -8b, and \text{\footnotesize $ 17}}

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Key Concepts

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

Polynomials

Polynomials are fundamental expressions in algebra that consist of variables, coefficients, and exponents, combined using addition, subtraction, and multiplication. Imagine polynomials as mathematical phrases. Each phrase is built from smaller mathematical words (terms), connected by addition or subtraction. For example, consider the polynomial expression:
-a^2b + 4a^2 - 8b + 17.
Here, the whole expression forms the polynomial.

Terms of a Polynomial

The terms of a polynomial are the building blocks that make up the entire expression. Each term is a distinct part of the polynomial separated by addition (+) or subtraction (-). In the polynomial -a^2b + 4a^2 - 8b + 17, the individual terms are:

-a^2b
4a^2
-8b
17

These terms are the separate entities that, when combined, form the whole polynomial.

Variables and Coefficients

Within each term of a polynomial, you'll find variables and coefficients. Variables are the letters (like a, b in our example) that can take different values. Coefficients are the numbers multiplying these variables. In the term -a^2b:

-1 is the coefficient (note the negative sign)
a^2 and b are variables

Similarly, in 4a^2:

4 is the coefficient
a^2 is the variable part

Recognizing these components within each term helps in manipulating and simplifying polynomials.

Expression Breaking

Breaking a polynomial expression involves separating it into its individual terms. This step allows us to analyze and work with each term separately. For our example -a^2b + 4a^2 - 8b + 17, we break it down as follows:
1. Start with the entire polynomial.
2. Identify each part of the polynomial that is added or subtracted.
3. Write down each term individually:

-a^2b
4a^2
-8b
17

This methodical process creates clarity and simplifies further operations, such as solving or factoring polynomials.

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91影视

Identify the terms of each polynomial. $$ -a^{2} b+4 a^{2}-8 b+17 $$

Short Answer

Step by step solution

Understand the Polynomial Structure

Identify the Terms by Breaking the Polynomial

List All Identified Terms

Key Concepts

Polynomials

Terms of a Polynomial

Variables and Coefficients

Expression Breaking

One App. One Place for Learning.

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