Answer The Questions About The Following Polynomial:$-x^2+\frac{1}{6}$1. The Expression Represents A { \square$}$ Polynomial With { \square$}$ Terms.2. The Constant Term Is { \square$}$.3. The Leading Term Is

In this article, we will delve into the world of polynomials and answer the questions related to the given expression: $-x^2+\frac{1}{6}$. We will explore the characteristics of this polynomial, including its degree, number of terms, constant term, and leading term.

Degree and Number of Terms

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 the given expression, $-x^2+\frac{1}{6}$, the highest power of the variable $x$ is 2. Therefore, the degree of this polynomial is 2.

The number of terms in a polynomial is the number of individual terms that make up the expression. In the given expression, there are two terms: $-x^2$ and $\frac{1}{6}$. Therefore, the expression represents a quadratic polynomial with two terms.

Constant Term

The constant term of a polynomial is the term that does not contain the variable. In the given expression, the constant term is $\frac{1}{6}$.

Leading Term

The leading term of a polynomial is the term with the highest degree. In the given expression, the leading term is $-x^2$.

Properties of the Polynomial

Now that we have identified the degree, number of terms, constant term, and leading term of the given polynomial, let's explore some of its properties.

• Even degree: Since the degree of the polynomial is 2, which is an even number, the polynomial is an even-degree polynomial.
• Negative leading coefficient: The leading coefficient of the polynomial is -1, which is a negative number. This means that the polynomial will have a downward-opening parabola.
• Constant term greater than zero: The constant term of the polynomial is $\frac{1}{6}$, which is greater than zero. This means that the polynomial will have a minimum value.

Graph of the Polynomial

The graph of the polynomial $-x^2+\frac{1}{6}$ is a downward-opening parabola with a minimum value at $x=0$. The parabola intersects the y-axis at $\frac{1}{6}$ and has a vertex at $(0, \frac{1}{6})$.

Solving the Polynomial

To solve the polynomial $-x^2+\frac{1}{6}=0$, we can set the expression equal to zero and solve for $x$. This gives us:

$$-x^2+\frac{1}{6}=0$$

Subtracting $\frac{1}{6}$ from both sides gives us:

$$-x^2=-\frac{1}{6}$$

Multiplying both sides by -1 gives us:

$$x^2=\frac{1}{6}$$

Taking the square root of both sides gives us:

$$x=\pm\sqrt{\frac{1}{6}}$$

Therefore, the solutions to the polynomial are $x=\sqrt{\frac{1}{6}}$ and $x=-\sqrt{\frac{1}{6}}$.

Conclusion

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In this article, we will answer some of the most frequently asked questions about the polynomial $-x^2+\frac{1}{6}$. Whether you are a student, a teacher, or simply someone interested in mathematics, this article will provide you with the information you need to understand this polynomial.

Q: What is the degree of the polynomial $-x^2+\frac{1}{6}$?

A: The degree of the polynomial $-x^2+\frac{1}{6}$ is 2. This is because the highest power of the variable $x$ in the polynomial is 2.

Q: How many terms are in the polynomial $-x^2+\frac{1}{6}$?

A: There are two terms in the polynomial $-x^2+\frac{1}{6}$. The first term is $-x^2$ and the second term is $\frac{1}{6}$.

Q: What is the constant term of the polynomial $-x^2+\frac{1}{6}$?

A: The constant term of the polynomial $-x^2+\frac{1}{6}$ is $\frac{1}{6}$. This is the term that does not contain the variable $x$.

Q: What is the leading term of the polynomial $-x^2+\frac{1}{6}$?

A: The leading term of the polynomial $-x^2+\frac{1}{6}$ is $-x^2$. This is the term with the highest degree in the polynomial.

Q: Is the polynomial $-x^2+\frac{1}{6}$ an even-degree polynomial?

A: Yes, the polynomial $-x^2+\frac{1}{6}$ is an even-degree polynomial. This is because the degree of the polynomial is 2, which is an even number.

Q: Will the graph of the polynomial $-x^2+\frac{1}{6}$ be a downward-opening parabola?

A: Yes, the graph of the polynomial $-x^2+\frac{1}{6}$ will be a downward-opening parabola. This is because the leading coefficient of the polynomial is -1, which is a negative number.

Q: What is the minimum value of the polynomial $-x^2+\frac{1}{6}$?

A: The minimum value of the polynomial $-x^2+\frac{1}{6}$ is $\frac{1}{6}$. This is the value of the polynomial at $x=0$.

Q: How do I solve the polynomial equation $-x^2+\frac{1}{6}=0$?

A: To solve the polynomial equation $-x^2+\frac{1}{6}=0$, you can set the expression equal to zero and solve for $x$. This gives you:

$$-x^2+\frac{1}{6}=0$$

Subtracting $\frac{1}{6}$ from both sides gives you:

$$-x^2=-\frac{1}{6}$$

Multiplying both sides by -1 gives you:

$$x^2=\frac{1}{6}$$

Taking the square root of both sides gives you:

$$x=\pm\sqrt{\frac{1}{6}}$$

Therefore, the solutions to the polynomial equation are $x=\sqrt{\frac{1}{6}}$ and $x=-\sqrt{\frac{1}{6}}$.

Conclusion

In this article, we have answered some of the most frequently asked questions about the polynomial $-x^2+\frac{1}{6}$. We have discussed the degree, number of terms, constant term, and leading term of the polynomial, as well as some of its properties. We have also provided solutions to the polynomial equation. Whether you are a student, a teacher, or simply someone interested in mathematics, this article will provide you with the information you need to understand this polynomial.