Which Of The Following Polynomials Is Written In Standard Form?A. Y = 7 X 3 − 5 X 2 + 2 X + 7 Y = 7x^3 - 5x^2 + 2x + 7 Y = 7 X 3 − 5 X 2 + 2 X + 7 B. Y = 8 − 4 X 2 + 3 X + 2 X 3 Y = 8 - 4x^2 + 3x + 2x^3 Y = 8 − 4 X 2 + 3 X + 2 X 3 C. Y = 1 Y = 1 Y = 1

Introduction

In mathematics, polynomials are algebraic expressions consisting of variables and coefficients combined using only addition, subtraction, and multiplication. The standard form of a polynomial is a crucial concept in algebra, and it plays a significant role in various mathematical operations, such as addition, subtraction, and multiplication of polynomials. In this article, we will explore the concept of standard form and determine which of the given polynomials is written in standard form.

What is Standard Form?

The standard form of a polynomial is a way of writing a polynomial expression in a specific order, with the terms arranged in descending order of their exponents. In other words, the term with the highest exponent comes first, followed by the term with the next highest exponent, and so on. The standard form of a polynomial is usually written with the variable (in this case, x) as the first term, followed by the coefficients and exponents.

Example of Standard Form

A polynomial in standard form can be written as:

y = ax^n + bx^(n-1) + cx^(n-2) + ... + k

Where:

• a, b, c, ... are coefficients
• n is the exponent of the variable x
• k is the constant term

Analyzing the Given Polynomials

Now, let's analyze the given polynomials and determine which one is written in standard form.

Polynomial A: $y = 7x^3 - 5x^2 + 2x + 7$

This polynomial has four terms, with the highest exponent being 3. The terms are arranged in descending order of their exponents, making it a candidate for standard form.

Polynomial B: $y = 8 - 4x^2 + 3x + 2x^3$

This polynomial also has four terms, but the highest exponent is 3. However, the terms are not arranged in descending order of their exponents, making it not a candidate for standard form.

Polynomial C: $y = 1$

This polynomial has only one term, which is a constant. It does not have any variable or exponent, making it not a candidate for standard form.

Conclusion

Based on the analysis, Polynomial A is the only one that is written in standard form. The terms are arranged in descending order of their exponents, making it a valid candidate for standard form.

Standard Form of Polynomials: Key Takeaways

• The standard form of a polynomial is a way of writing a polynomial expression in a specific order, with the terms arranged in descending order of their exponents.
• The term with the highest exponent comes first, followed by the term with the next highest exponent, and so on.
• The standard form of a polynomial is usually written with the variable (in this case, x) as the first term, followed by the coefficients and exponents.

Real-World Applications of Standard Form

The concept of standard form is crucial in various mathematical operations, such as:

• Addition and subtraction of polynomials
• Multiplication of polynomials
• Factoring of polynomials
• Graphing of polynomial functions

In conclusion, the standard form of a polynomial is a fundamental concept in algebra, and it plays a significant role in various mathematical operations. By understanding the concept of standard form, we can perform mathematical operations more efficiently and accurately.

Common Mistakes to Avoid

When writing a polynomial in standard form, it's essential to avoid common mistakes, such as:

• Not arranging the terms in descending order of their exponents
• Not including the variable (in this case, x) as the first term
• Not including the coefficients and exponents in the correct order

By avoiding these common mistakes, we can ensure that our polynomial is written in standard form and can be used for various mathematical operations.

Final Thoughts

Frequently Asked Questions

In this article, we will address some of the most frequently asked questions about the standard form of polynomials.

Q: What is the standard form of a polynomial?

A: The standard form of a polynomial is a way of writing a polynomial expression in a specific order, with the terms arranged in descending order of their exponents.

Q: Why is the standard form of a polynomial important?

A: The standard form of a polynomial is important because it allows us to perform mathematical operations, such as addition, subtraction, and multiplication, more efficiently and accurately.

Q: How do I write a polynomial in standard form?

A: To write a polynomial in standard form, you need to arrange the terms in descending order of their exponents. The term with the highest exponent comes first, followed by the term with the next highest exponent, and so on.

Q: What are some common mistakes to avoid when writing a polynomial in standard form?

A: Some common mistakes to avoid when writing a polynomial in standard form include:

• Not arranging the terms in descending order of their exponents
• Not including the variable (in this case, x) as the first term
• Not including the coefficients and exponents in the correct order

Q: Can you give an example of a polynomial in standard form?

A: Yes, here is an example of a polynomial in standard form:

y = 3x^2 - 2x + 1

In this example, the terms are arranged in descending order of their exponents, with the term with the highest exponent (2) coming first.

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

A: To add or subtract polynomials in standard form, you need to combine like terms. Like terms are terms that have the same variable and exponent. For example:

y = 2x^2 + 3x + 1 y = x^2 - 2x - 1

To add these two polynomials, you would combine like terms:

y = (2x^2 + x^2) + (3x - 2x) + (1 - 1) y = 3x^2 + x + 0

Q: How do I multiply polynomials in standard form?

A: To multiply polynomials in standard form, you need to use the distributive property. The distributive property states that for any numbers a, b, and c:

a(b + c) = ab + ac

For example:

y = (2x^2 + 3x + 1)(x + 2)

To multiply these two polynomials, you would use the distributive property:

y = 2x^2(x) + 2x^2(2) + 3x(x) + 3x(2) + 1(x) + 1(2) y = 2x^3 + 4x^2 + 3x^2 + 6x + x + 2 y = 2x^3 + 7x^2 + 7x + 2

Q: Can you give an example of a polynomial that is not in standard form?

A: Yes, here is an example of a polynomial that is not in standard form:

y = x^2 + 3x - 1 + 2x^3

In this example, the terms are not arranged in descending order of their exponents, making it not a candidate for standard form.

Conclusion

In conclusion, the standard form of a polynomial is a crucial concept in algebra, and it plays a significant role in various mathematical operations. By understanding the concept of standard form, we can perform mathematical operations more efficiently and accurately. Remember to always arrange the terms in descending order of their exponents and include the variable (in this case, x) as the first term. With practice and patience, you'll become proficient in writing polynomials in standard form.