Identifying the Degree of the Given Polynomial

Oct 13, 2025 by miodeya.com 70 views

Introduction

In algebra, the degree of a polynomial is a crucial concept that helps us understand the properties and behavior of the polynomial. It is defined as the highest power of the variable in the polynomial. In this article, we will explore the concept of the degree of a polynomial and apply it to the given expression $$6x^5 - 4x^2 + 2x^3 - 3 + x$$.

What is the Degree of a Polynomial?

The degree of a polynomial is the highest power of the variable in the polynomial. For example, in the polynomial $$3x^2 + 2x - 1$$, the highest power of the variable x is 2, so the degree of the polynomial is 2.

Identifying the Degree of the Given Polynomial

To find the degree of the given polynomial $$6x^5 - 4x^2 + 2x^3 - 3 + x$$, we need to identify the highest power of the variable x. We can do this by looking at each term in the polynomial and identifying the power of x in each term.

Term 1: $$6x^5$$

The power of x in the first term is 5.

Term 2: $$-4x^2$$

The power of x in the second term is 2.

Term 3: $$2x^3$$

The power of x in the third term is 3.

Term 4: $$-3$$

The power of x in the fourth term is 0, since there is no x in this term.

Term 5: $$x$$

The power of x in the fifth term is 1.

Comparing the Powers of x

Now that we have identified the power of x in each term, we can compare them to find the highest power of x. The highest power of x is 5, which is the power of x in the first term.

Conclusion

Therefore, the degree of the polynomial $$6x^5 - 4x^2 + 2x^3 - 3 + x$$ is 5.

Answer

The correct answer is B. 5.

Frequently Asked Questions

What is the degree of a polynomial? The degree of a polynomial is the highest power of the variable in the polynomial.
How do I find the degree of a polynomial? To find the degree of a polynomial, you need to identify the highest power of the variable in each term and compare them to find the highest power.
What is the degree of the polynomial $$6x^5 - 4x^2 + 2x^3 - 3 + x$$? The degree of the polynomial $$6x^5 - 4x^2 + 2x^3 - 3 + x$$ is 5.

References

[1] Algebra, 2nd edition, by Michael Artin
[2] Calculus, 3rd edition, by Michael Spivak
[3] Polynomial, Wikipedia

Conclusion

In conclusion, the degree of a polynomial is a crucial concept in algebra that helps us understand the properties and behavior of the polynomial. By identifying the highest power of the variable in each term and comparing them, we can find the degree of a polynomial. In this article, we applied this concept to the given expression $$6x^5 - 4x^2 + 2x^3 - 3 + x$$ and found that the degree of the polynomial is 5.

Introduction

In our previous article, we discussed the concept of the degree of a polynomial and applied it to the given expression $$6x^5 - 4x^2 + 2x^3 - 3 + x$$. In this article, we will answer some frequently asked questions related to the degree of a polynomial.

Q&A

Q1: What is the degree of a polynomial?

A1: The degree of a polynomial is the highest power of the variable in the polynomial.

Q2: How do I find the degree of a polynomial?

A2: To find the degree of a polynomial, you need to identify the highest power of the variable in each term and compare them to find the highest power.

Q3: What is the degree of a polynomial with no variable?

A3: The degree of a polynomial with no variable is 0.

Q4: What is the degree of a polynomial with only one term?

A4: The degree of a polynomial with only one term is the power of the variable in that term.

Q5: Can a polynomial have a negative degree?

A5: No, a polynomial cannot have a negative degree. The degree of a polynomial is always a non-negative integer.

Q6: How do I determine the degree of a polynomial with multiple variables?

A6: To determine the degree of a polynomial with multiple variables, you need to identify the highest power of each variable and multiply them together. The resulting product is the degree of the polynomial.

Q7: What is the degree of a polynomial with a fractional exponent?

A7: The degree of a polynomial with a fractional exponent is the denominator of the exponent.

Q8: Can a polynomial have a degree of 0?

A8: Yes, a polynomial can have a degree of 0. This occurs when the polynomial has no variable or when the variable has a power of 0.

Q9: How do I determine the degree of a polynomial with a negative exponent?

A9: To determine the degree of a polynomial with a negative exponent, you need to take the reciprocal of the exponent and multiply it by the coefficient of the term. The resulting product is the degree of the polynomial.

Q10: What is the degree of a polynomial with a complex exponent?

A10: The degree of a polynomial with a complex exponent is the magnitude of the exponent.

Examples

Example 1: Find the degree of the polynomial $$2x^3 + 3x^2 - 4x + 1$$.

To find the degree of the polynomial, we need to identify the highest power of the variable x. In this case, the highest power of x is 3, so the degree of the polynomial is 3.

Example 2: Find the degree of the polynomial $$x^2 + 2x - 3$$.

To find the degree of the polynomial, we need to identify the highest power of the variable x. In this case, the highest power of x is 2, so the degree of the polynomial is 2.

Example 3: Find the degree of the polynomial $$x^3 - 2x^2 + 3x - 1$$.

To find the degree of the polynomial, we need to identify the highest power of the variable x. In this case, the highest power of x is 3, so the degree of the polynomial is 3.

Conclusion

In conclusion, the degree of a polynomial is a crucial concept in algebra that helps us understand the properties and behavior of the polynomial. By answering some frequently asked questions related to the degree of a polynomial, we have provided a comprehensive overview of this concept. We hope that this article has been helpful in clarifying any doubts you may have had about the degree of a polynomial.

References

[1] Algebra, 2nd edition, by Michael Artin
[2] Calculus, 3rd edition, by Michael Spivak
[3] Polynomial, Wikipedia