Which Algebraic Expression Is A Polynomial With A Degree Of $4$?A. $5x^4 + \sqrt{4x}$ B. $x^5 - 6x^4 + 14x^3 + X^2$ C. \$9x^4 - X^3 - \frac{x}{5}$[/tex\] D. $2x^4 - 6x^4 + \frac{14}{x}$

In mathematics, polynomials are algebraic expressions consisting of variables and coefficients combined using only addition, subtraction, and multiplication. The degree of a polynomial is determined by the highest power of the variable present in the expression. In this article, we will explore which of the given algebraic expressions is a polynomial with a degree of 4.

Understanding Polynomials and Their Degrees

A polynomial is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. The degree of a polynomial is determined by the highest power of the variable present in the expression. For example, in the expression $3x^2 + 2x - 1$, the highest power of the variable $x$ is 2, so the degree of this polynomial is 2.

Analyzing the Given Algebraic Expressions

We are given four algebraic expressions, and we need to determine which one is a polynomial with a degree of 4.

A. $5x^4 + \sqrt{4x}$

This expression consists of two terms: $5x^4$ and $\sqrt{4x}$. The first term has a degree of 4, as it is $x$ raised to the power of 4. However, the second term, $\sqrt{4x}$, is not a polynomial because it involves a square root, which is not a polynomial operation. Therefore, this expression is not a polynomial with a degree of 4.

B. $x^5 - 6x^4 + 14x^3 + x^2$

This expression consists of four terms: $x^5$, $-6x^4$, $14x^3$, and $x^2$. The highest power of the variable $x$ in this expression is 5, which is the degree of the polynomial. Therefore, this expression is a polynomial with a degree of 5, not 4.

C. $9x^4 - x^3 - \frac{x}{5}$

This expression consists of three terms: $9x^4$, $-x^3$, and $-\frac{x}{5}$. The highest power of the variable $x$ in this expression is 4, which is the degree of the polynomial. Therefore, this expression is a polynomial with a degree of 4.

D. $2x^4 - 6x^4 + \frac{14}{x}$

This expression consists of three terms: $2x^4$, $-6x^4$, and $\frac{14}{x}$. The highest power of the variable $x$ in this expression is 4, which is the degree of the polynomial. However, the third term, $\frac{14}{x}$, is not a polynomial because it involves a fraction, which is not a polynomial operation. Therefore, this expression is not a polynomial with a degree of 4.

Conclusion

In conclusion, the algebraic expression that is a polynomial with a degree of 4 is C. $9x^4 - x^3 - \frac{x}{5}$. This expression consists of three terms, with the highest power of the variable $x$ being 4, which is the degree of the polynomial.

Key Takeaways

• A polynomial is an algebraic expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication.
• The degree of a polynomial is determined by the highest power of the variable present in the expression.
• To determine the degree of a polynomial, we need to identify the highest power of the variable present in the expression.
• A polynomial with a degree of 4 is an expression with the highest power of the variable being 4.

Frequently Asked Questions

Q: What is a polynomial?

A: A polynomial is an algebraic expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication.

Q: How is the degree of a polynomial determined?

A: The degree of a polynomial is determined by the highest power of the variable present in the expression.

Q: What is the degree of the polynomial $3x^2 + 2x - 1$?

A: The degree of the polynomial $3x^2 + 2x - 1$ is 2, as the highest power of the variable $x$ is 2.

Q: What is the degree of the polynomial $x^5 - 6x^4 + 14x^3 + x^2$?

A: The degree of the polynomial $x^5 - 6x^4 + 14x^3 + x^2$ is 5, as the highest power of the variable $x$ is 5.

Q: What is the degree of the polynomial $9x^4 - x^3 - \frac{x}{5}$?

A: The degree of the polynomial $9x^4 - x^3 - \frac{x}{5}$ is 4, as the highest power of the variable $x$ is 4.

Q: What is the degree of the polynomial $2x^4 - 6x^4 + \frac{14}{x}$?

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In this article, we will continue to explore the concept of polynomial degree and answer some frequently asked questions.

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

A: A polynomial expression is an expression that consists of variables and coefficients combined using only addition, subtraction, and multiplication. On the other hand, a non-polynomial expression is an expression that involves operations such as division, square roots, or other non-polynomial operations.

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

A: To determine the degree of a polynomial, you need to identify the highest power of the variable present in the expression. For example, in the expression $3x^2 + 2x - 1$, the highest power of the variable $x$ is 2, so the degree of this polynomial is 2.

Q: What is the degree of the polynomial $x^3 + 2x^2 - 3x + 1$?

A: The degree of the polynomial $x^3 + 2x^2 - 3x + 1$ is 3, as the highest power of the variable $x$ is 3.

Q: What is the degree of the polynomial $2x^4 - 3x^2 + 1$?

A: The degree of the polynomial $2x^4 - 3x^2 + 1$ is 4, as the highest power of the variable $x$ is 4.

Q: Can a polynomial have a negative degree?

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

Q: What is the degree of the polynomial $x^{-2} + 2x^{-1} - 3$?

A: The degree of the polynomial $x^{-2} + 2x^{-1} - 3$ is -2, as the highest power of the variable $x$ is -2. However, this is not a valid polynomial degree, as the degree of a polynomial must be a non-negative integer.

Q: Can a polynomial have a fractional degree?

A: No, a polynomial cannot have a fractional degree. The degree of a polynomial must be a non-negative integer.

Q: What is the degree of the polynomial $x^{\frac{1}{2}} + 2x^{\frac{1}{3}} - 3$?

A: The degree of the polynomial $x^{\frac{1}{2}} + 2x^{\frac{1}{3}} - 3$ is not a valid polynomial degree, as the expression involves fractional powers of the variable $x$.

Q: How do I simplify a polynomial expression?

A: To simplify a polynomial expression, you need to combine like terms and eliminate any unnecessary operations. For example, in the expression $3x^2 + 2x^2 - 3x + 1$, you can combine the like terms $3x^2$ and $2x^2$ to get $5x^2$.

Q: What is the simplified form of the polynomial expression $2x^3 + 3x^2 - 4x + 1$?

A: The simplified form of the polynomial expression $2x^3 + 3x^2 - 4x + 1$ is $2x^3 + 3x^2 - 4x + 1$, as there are no like terms to combine.

Q: Can a polynomial expression be factored?

A: Yes, a polynomial expression can be factored using various techniques such as factoring out common factors, using the distributive property, or using the quadratic formula.

Q: How do I factor a polynomial expression?

A: To factor a polynomial expression, you need to identify any common factors and factor them out, or use the distributive property to factor the expression. For example, in the expression $6x^2 + 12x + 6$, you can factor out the common factor $6$ to get $6(x^2 + 2x + 1)$.

Q: What is the factored form of the polynomial expression $x^2 + 4x + 4$?

A: The factored form of the polynomial expression $x^2 + 4x + 4$ is $(x + 2)^2$, as it can be factored using the perfect square trinomial formula.

Q: Can a polynomial expression be solved using algebraic methods?

A: Yes, a polynomial expression can be solved using algebraic methods such as factoring, using the quadratic formula, or using other algebraic techniques.

Q: How do I solve a polynomial expression using algebraic methods?

A: To solve a polynomial expression using algebraic methods, you need to identify the type of polynomial and use the appropriate algebraic technique to solve it. For example, in the expression $x^2 + 4x + 4 = 0$, you can use the quadratic formula to solve for $x$.

Q: What is the solution to the polynomial expression $x^2 + 4x + 4 = 0$?

A: The solution to the polynomial expression $x^2 + 4x + 4 = 0$ is $x = -2$, as it can be solved using the quadratic formula.

Conclusion

In conclusion, the degree of a polynomial is an important concept in algebra that determines the highest power of the variable present in the expression. By understanding the degree of a polynomial, you can simplify and solve polynomial expressions using various algebraic techniques.