Choose The Correct Classification Of $5x + 3x^4 - 7x^3 + 10$ By Number Of Terms And By Degree.A. Third Degree Polynomial B. Fourth Degree Polynomial C. Sixth Degree Polynomial D. First Degree Binomial

Polynomials are algebraic expressions consisting of variables and coefficients combined using only addition, subtraction, and multiplication. When it comes to classifying polynomials, two key characteristics are considered: the number of terms and the degree of the polynomial. In this article, we will explore how to determine the correct classification of a given polynomial based on these two factors.

Understanding the Number of Terms

The number of terms in a polynomial refers to the count of individual components or parts that make up the expression. For example, in the polynomial $5x + 3x^4 - 7x^3 + 10$, there are four distinct terms: $5x$, $3x^4$, $-7x^3$, and $10$. This polynomial has four terms.

Understanding the Degree of a Polynomial

The degree of a polynomial is determined by the highest power or exponent of the variable (in this case, $x$) present in any of the terms. In the polynomial $5x + 3x^4 - 7x^3 + 10$, the highest power of $x$ is $4$, which is the exponent in the term $3x^4$. Therefore, the degree of this polynomial is $4$.

Classifying Polynomials by Degree

Polynomials can be classified based on their degree as follows:

• Zero-degree polynomial: A polynomial with no variable, such as $5$.
• First-degree polynomial: A polynomial with a degree of $1$, such as $2x$ or $3x + 4$.
• Second-degree polynomial: A polynomial with a degree of $2$, such as $x^2 + 3x - 4$.
• Third-degree polynomial: A polynomial with a degree of $3$, such as $x^3 + 2x^2 - 3x + 1$.
• Fourth-degree polynomial: A polynomial with a degree of $4$, such as $x^4 + 3x^3 - 2x^2 + x + 1$.
• Higher-degree polynomial: A polynomial with a degree greater than $4$, such as $x^5 + 2x^4 - 3x^3 + x^2 + 1$.

Classifying Polynomials by Number of Terms

Polynomials can also be classified based on the number of terms as follows:

• Monomial: A polynomial with only one term, such as $5x$ or $3x^4$.
• Binomial: A polynomial with two terms, such as $3x + 4$ or $x^2 - 2x$.
• Trinomial: A polynomial with three terms, such as $x^2 + 3x - 4$ or $2x^3 - 3x^2 + x$.
• Polynomial with more than three terms: A polynomial with four or more terms, such as $x^4 + 3x^3 - 2x^2 + x + 1$.

Applying the Classification Criteria to the Given Polynomial

Now that we have a clear understanding of the number of terms and degree of a polynomial, let's apply these criteria to the given polynomial $5x + 3x^4 - 7x^3 + 10$.

• Number of terms: The polynomial has four distinct terms, making it a polynomial with more than three terms.
• Degree: The highest power of $x$ in the polynomial is $4$, making it a fourth-degree polynomial.

Conclusion

In conclusion, the given polynomial $5x + 3x^4 - 7x^3 + 10$ is a fourth-degree polynomial with four terms. This classification is based on the highest power of the variable $x$ present in the polynomial, which is $4$, and the count of individual components or parts that make up the expression, which is four.

Answer

The correct classification of the polynomial $5x + 3x^4 - 7x^3 + 10$ by number of terms and by degree is:

• Number of terms: Polynomial with more than three terms
• Degree: Fourth-degree polynomial

The correct answer is:

In the previous article, we explored the concept of classifying polynomials based on the number of terms and degree. Here, we will address some frequently asked questions related to this topic.

Q: What is the difference between a polynomial and a non-polynomial expression?

A polynomial expression is an algebraic expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. On the other hand, a non-polynomial expression may include other operations such as division, roots, or fractions.

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

To determine the degree of a polynomial, you need to identify the highest power or exponent of the variable (in this case, x) present in any of the terms. For example, in the polynomial 3x^4 + 2x^3 - x^2 + x, the highest power of x is 4, making it a fourth-degree polynomial.

Q: What is the difference between a monomial, binomial, and trinomial?

A monomial is a polynomial with only one term, such as 3x or 2x^2. A binomial is a polynomial with two terms, such as 3x + 2 or x^2 - 2x. A trinomial is a polynomial with three terms, such as x^2 + 3x - 4 or 2x^3 - 3x^2 + x.

Q: Can a polynomial have more than three terms?

Yes, a polynomial can have more than three terms. For example, the polynomial 3x^4 + 2x^3 - x^2 + x + 1 has five terms.

Q: How do I classify a polynomial with a negative exponent?

A polynomial with a negative exponent is still classified based on the highest power of the variable. For example, in the polynomial 3x^(-4) + 2x^3 - x^2 + x, the highest power of x is 3, making it a third-degree polynomial.

Q: Can a polynomial have a zero degree?

Yes, a polynomial can have a zero degree. This occurs when the polynomial has no variable, such as 5 or 3.

Q: How do I classify a polynomial with a variable raised to a fractional exponent?

A polynomial with a variable raised to a fractional exponent is still classified based on the highest power of the variable. For example, in the polynomial 3x^(1/2) + 2x^3 - x^2 + x, the highest power of x is 3, making it a third-degree polynomial.

Q: Can a polynomial have a variable raised to a negative fractional exponent?

Yes, a polynomial can have a variable raised to a negative fractional exponent. For example, in the polynomial 3x^(-1/2) + 2x^3 - x^2 + x, the highest power of x is 3, making it a third-degree polynomial.

Q: How do I classify a polynomial with a variable raised to a negative integer exponent?

A polynomial with a variable raised to a negative integer exponent is still classified based on the highest power of the variable. For example, in the polynomial 3x^(-4) + 2x^3 - x^2 + x, the highest power of x is 3, making it a third-degree polynomial.

Q: Can a polynomial have a variable raised to a negative integer exponent and a fractional exponent?

Yes, a polynomial can have a variable raised to a negative integer exponent and a fractional exponent. For example, in the polynomial 3x^(-4/2) + 2x^3 - x^2 + x, the highest power of x is 3, making it a third-degree polynomial.

Conclusion

In conclusion, classifying polynomials based on the number of terms and degree is a crucial concept in algebra. By understanding the characteristics of polynomials, you can accurately classify them and solve various mathematical problems.