Finding the Coefficient of the Term of Degree 7

Oct 13, 2025 by miodeya.com 108 views

Introduction

In algebra, a polynomial is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. The degree of a polynomial is the highest power of the variable in any of its terms. In this article, we will focus on finding the coefficient of the term of degree 7 in a given polynomial.

Understanding Polynomials

A polynomial can be written in the form:

a_n x^n + a_(n-1) x^(n-1) + ... + a_1 x + a_0

where a_n, a_(n-1), ..., a_1, a_0 are constants, and x is the variable. The degree of the polynomial is n.

The Given Polynomial

The given polynomial is:

2x^6 + 2 - 4x^2 + 5x^7 - 4x

Finding the Coefficient of the Term of Degree 7

To find the coefficient of the term of degree 7, we need to identify the term with the highest power of x, which is 7 in this case. The term with degree 7 is 5x^7.

Identifying the Coefficient

The coefficient of a term is the constant that is multiplied by the variable. In this case, the coefficient of the term 5x^7 is 5.

Conclusion

Therefore, the coefficient of the term of degree 7 in the given polynomial is 5.

Final Answer

The final answer is C. 5.

Discussion

The coefficient of a term in a polynomial is an important concept in algebra. It is used to determine the behavior of the polynomial, such as its roots and its graph. In this article, we have seen how to find the coefficient of a term of a given degree in a polynomial.

Example Problems

Find the coefficient of the term of degree 5 in the polynomial 3x^5 + 2x^2 - 4x + 1.
Find the coefficient of the term of degree 3 in the polynomial 2x^3 - 3x^2 + 4x - 1.

Solutions

The coefficient of the term of degree 5 in the polynomial 3x^5 + 2x^2 - 4x + 1 is 3.
The coefficient of the term of degree 3 in the polynomial 2x^3 - 3x^2 + 4x - 1 is 2.

Conclusion

In conclusion, finding the coefficient of a term of a given degree in a polynomial is an important concept in algebra. It is used to determine the behavior of the polynomial, such as its roots and its graph. We have seen how to find the coefficient of a term of a given degree in a polynomial, and we have also seen some example problems and their solutions.

Introduction

In our previous article, we discussed how to find the coefficient of a term of a given degree in a polynomial. In this article, we will answer some frequently asked questions (FAQs) about coefficients of terms in polynomials.

Q&A

Q1: What is the coefficient of a term in a polynomial?

A1: The coefficient of a term in a polynomial is the constant that is multiplied by the variable. For example, in the term 5x^7, the coefficient is 5.

Q2: How do I find the coefficient of a term in a polynomial?

A2: To find the coefficient of a term in a polynomial, you need to identify the term with the given degree and multiply the constant by the variable. For example, to find the coefficient of the term of degree 7 in the polynomial 2x^6 + 2 - 4x^2 + 5x^7 - 4x, you would identify the term 5x^7 and multiply the constant 5 by the variable x^7.

Q3: What is the difference between the coefficient and the constant in a polynomial?

A3: The coefficient and the constant are related but distinct concepts in a polynomial. The coefficient is the constant that is multiplied by the variable, while the constant is the number that is not multiplied by a variable. For example, in the term 5x^7, the coefficient is 5 and the constant is 5x^7.

Q4: Can a term in a polynomial have a coefficient of zero?

A4: Yes, a term in a polynomial can have a coefficient of zero. For example, in the polynomial 2x^6 + 2 - 4x^2 + 0x^7 - 4x, the term 0x^7 has a coefficient of zero.

Q5: How do I determine the degree of a polynomial?

A5: To determine the degree of a polynomial, you need to identify the term with the highest power of the variable. For example, in the polynomial 2x^6 + 2 - 4x^2 + 5x^7 - 4x, the term 5x^7 has the highest power of x, so the degree of the polynomial is 7.

Q6: Can a polynomial have multiple terms with the same degree?

A6: Yes, a polynomial can have multiple terms with the same degree. For example, in the polynomial 2x^6 + 2 - 4x^2 + 5x^7 - 4x + 3x^7, the terms 5x^7 and 3x^7 have the same degree, which is 7.

Q7: How do I simplify a polynomial with multiple terms of the same degree?

A7: To simplify a polynomial with multiple terms of the same degree, you need to combine the coefficients of the terms with the same degree. For example, in the polynomial 2x^6 + 2 - 4x^2 + 5x^7 - 4x + 3x^7, you would combine the coefficients of the terms 5x^7 and 3x^7 to get 8x^7.

Q8: Can a polynomial have a term with a negative coefficient?

A8: Yes, a polynomial can have a term with a negative coefficient. For example, in the polynomial 2x^6 + 2 - 4x^2 - 5x^7 - 4x, the term -5x^7 has a negative coefficient.

Q9: How do I determine the behavior of a polynomial?

A9: To determine the behavior of a polynomial, you need to examine its degree and the signs of its coefficients. For example, a polynomial with an even degree and positive coefficients will have a minimum value, while a polynomial with an odd degree and negative coefficients will have a maximum value.

Q10: Can a polynomial have a term with a fractional coefficient?

A10: Yes, a polynomial can have a term with a fractional coefficient. For example, in the polynomial 2x^6 + 2 - 4x^2 + 5/2x^7 - 4x, the term 5/2x^7 has a fractional coefficient.

Conclusion

In conclusion, coefficients of terms in polynomials are an important concept in algebra. We have answered some frequently asked questions (FAQs) about coefficients of terms in polynomials, including how to find the coefficient of a term, the difference between the coefficient and the constant, and how to determine the degree of a polynomial.