Understanding the Rules of Standard Form

Oct 13, 2025 by windailys.com 28 views

Introduction

When creating a polynomial written in standard form, it is essential to understand the rules and requirements that govern the arrangement of terms. A polynomial in standard form is arranged in descending order of exponents, with the term having the highest exponent first. In this article, we will explore which terms could be used as the last term of the given expression to create a polynomial written in standard form.

Understanding the Rules of Standard Form

Before we dive into the options, let's review the rules of standard form for polynomials. A polynomial in standard form is arranged in descending order of exponents, with the term having the highest exponent first. This means that the term with the highest power of x and y should come first, followed by the term with the next highest power, and so on.

Examining the Given Expression

The given expression is:

-5x^2y^4 + 9x^3y^3

To create a polynomial written in standard form, we need to add a term that will be the last term in the expression. This term should have a lower exponent than the existing terms, but it should still be a polynomial term.

Option A: $x^5$

Option A is $x^5$ . If we add this term to the given expression, the resulting polynomial would be:

-5x^2y^4 + 9x^3y^3 + x^5

However, this term has a higher exponent than the existing terms, which means it would not be the last term in the standard form. Therefore, option A is not a valid choice.

Option B: $y^5$

Option B is $y^5$ . If we add this term to the given expression, the resulting polynomial would be:

-5x^2y^4 + 9x^3y^3 + y^5

In this case, the term $y^5$ has a lower exponent than the existing terms, which means it would be the last term in the standard form. Therefore, option B is a valid choice.

Option C: $-4x^4y^5$

Option C is $-4x^4y^5$ . If we add this term to the given expression, the resulting polynomial would be:

-5x^2y^4 + 9x^3y^3 - 4x^4y^5

However, this term has a higher exponent than the existing terms, which means it would not be the last term in the standard form. Therefore, option C is not a valid choice.

Conclusion

In conclusion, the only valid option for the last term of the expression to create a polynomial written in standard form is option B: $y^5$ . This term has a lower exponent than the existing terms, which means it would be the last term in the standard form.

Frequently Asked Questions

What is the standard form of a polynomial? The standard form of a polynomial is arranged in descending order of exponents, with the term having the highest exponent first.
How do I determine the last term of a polynomial in standard form? To determine the last term of a polynomial in standard form, you need to look for the term with the lowest exponent. This term should be added to the end of the expression.
What is the difference between a polynomial and a standard form polynomial? A polynomial is a mathematical expression consisting of variables and coefficients, while a standard form polynomial is a polynomial arranged in descending order of exponents.

Final Answer

The final answer is option B: $y^5$ .

Introduction

In our previous article, we explored which terms could be used as the last term of the given expression to create a polynomial written in standard form. In this article, we will answer some frequently asked questions about polynomials in standard form.

Q&A

Q: What is the standard form of a polynomial?

A: The standard form of a polynomial is arranged in descending order of exponents, with the term having the highest exponent first.

Q: How do I determine the last term of a polynomial in standard form?

A: To determine the last term of a polynomial in standard form, you need to look for the term with the lowest exponent. This term should be added to the end of the expression.

Q: What is the difference between a polynomial and a standard form polynomial?

A: A polynomial is a mathematical expression consisting of variables and coefficients, while a standard form polynomial is a polynomial arranged in descending order of exponents.

Q: Can I have multiple terms with the same exponent in a polynomial in standard form?

A: Yes, you can have multiple terms with the same exponent in a polynomial in standard form. However, the terms with the same exponent should be grouped together, with the term having the highest coefficient first.

Q: How do I add or subtract polynomials in standard form?

A: To add or subtract polynomials in standard form, you need to add or subtract the corresponding terms. For example, if you have two polynomials:

2x^2 + 3x + 1

and

x^2 + 2x - 1

You can add them by adding the corresponding terms:

(2x^2 + 3x + 1) + (x^2 + 2x - 1) = 3x^2 + 5x

Q: Can I have a polynomial with no terms?

A: Yes, you can have a polynomial with no terms. This is called the zero polynomial, and it is denoted by 0.

Q: How do I multiply polynomials in standard form?

A: To multiply polynomials in standard form, you need to multiply each term of the first polynomial by each term of the second polynomial, and then combine like terms.

Q: Can I have a polynomial with a negative exponent?

A: Yes, you can have a polynomial with a negative exponent. However, the term with a negative exponent should be written with a positive exponent and a negative coefficient.

Q: How do I divide polynomials in standard form?

A: To divide polynomials in standard form, you need to divide each term of the dividend by the divisor, and then combine like terms.

Conclusion

In conclusion, polynomials in standard form are an essential concept in algebra. By understanding the rules and requirements of standard form, you can perform various operations with polynomials, such as addition, subtraction, multiplication, and division.

Final Answer

The final answer is that polynomials in standard form are a fundamental concept in algebra, and understanding the rules and requirements of standard form is essential for performing various operations with polynomials.

Additional Resources

Final Tips

Always arrange polynomials in standard form before performing operations.
Use the distributive property to multiply polynomials.
Combine like terms when adding or subtracting polynomials.
Use long division to divide polynomials.
Practice, practice, practice!