Zeros Of The Polynomial P ( X ) = X 2 − 3 2 X + 4 P(x) = X^2 - 3\sqrt{2}x + 4 P ( X ) = X 2 − 3 2 ​ X + 4 A. 0 , − 3 0, -\sqrt{3} 0 , − 3 ​ B. 2 2 , 2 2\sqrt{2}, \sqrt{2} 2 2 ​ , 2 ​ C. 4 2 , − 2 4\sqrt{2}, -\sqrt{2} 4 2 ​ , − 2 ​ D. 2 , 2 \sqrt{2}, 2 2 ​ , 2

Introduction

In mathematics, a polynomial is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. The zeros of a polynomial are the values of the variable that make the polynomial equal to zero. In this article, we will focus on finding the zeros of a given polynomial, specifically the polynomial $P(x) = x^2 - 3\sqrt{2}x + 4$. We will use various techniques to find the zeros of this polynomial and explore the different options provided.

Understanding the Polynomial

The given polynomial is a quadratic polynomial, which means it has a degree of 2. The general form of a quadratic polynomial is $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. In this case, the polynomial is $P(x) = x^2 - 3\sqrt{2}x + 4$. To find the zeros of this polynomial, we need to set it equal to zero and solve for $x$.

Setting the Polynomial Equal to Zero

To find the zeros of the polynomial, we set it equal to zero: $P(x) = 0$. This gives us the equation $x^2 - 3\sqrt{2}x + 4 = 0$. We can now use various techniques to solve for $x$.

Factoring the Polynomial

One way to solve the quadratic equation is to factor the polynomial. We can try to find two numbers whose product is $4$ and whose sum is $-3\sqrt{2}$. These numbers are $-2\sqrt{2}$ and $-2\sqrt{2}$. Therefore, we can write the polynomial as $(x - 2\sqrt{2})(x - 2\sqrt{2}) = 0$. This gives us two possible solutions: $x = 2\sqrt{2}$ and $x = 2\sqrt{2}$.

Using the Quadratic Formula

Another way to solve the quadratic equation is to use the quadratic formula. The quadratic formula is given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. In this case, $a = 1$, $b = -3\sqrt{2}$, and $c = 4$. Plugging these values into the quadratic formula, we get $x = \frac{-(-3\sqrt{2}) \pm \sqrt{(-3\sqrt{2})^2 - 4(1)(4)}}{2(1)}$. Simplifying this expression, we get $x = \frac{3\sqrt{2} \pm \sqrt{18 - 16}}{2}$. This gives us two possible solutions: $x = \frac{3\sqrt{2} + \sqrt{2}}{2}$ and $x = \frac{3\sqrt{2} - \sqrt{2}}{2}$.

Simplifying the Solutions

We can simplify the solutions by combining like terms. The first solution becomes $x = \frac{4\sqrt{2}}{2}$, which simplifies to $x = 2\sqrt{2}$. The second solution becomes $x = \frac{2\sqrt{2}}{2}$, which simplifies to $x = \sqrt{2}$.

Conclusion

In conclusion, we have found the zeros of the polynomial $P(x) = x^2 - 3\sqrt{2}x + 4$. The zeros are $x = 2\sqrt{2}$ and $x = \sqrt{2}$. These values make the polynomial equal to zero, and they satisfy the equation $P(x) = 0$.

Answer Options

Now that we have found the zeros of the polynomial, we can compare them to the answer options provided. The answer options are:

A. $0, -\sqrt{3}$ B. $2\sqrt{2}, \sqrt{2}$ C. $4\sqrt{2}, -\sqrt{2}$ D. $\sqrt{2}, 2$

Comparing the zeros we found to the answer options, we can see that the correct answer is:

B. $2\sqrt{2}, \sqrt{2}$

This is the only option that matches the zeros we found.

Final Thoughts

In this article, we have used various techniques to find the zeros of a given polynomial. We have factored the polynomial and used the quadratic formula to solve for $x$. We have also simplified the solutions to find the final values of $x$. The zeros of the polynomial are $x = 2\sqrt{2}$ and $x = \sqrt{2}$. These values make the polynomial equal to zero, and they satisfy the equation $P(x) = 0$. We have compared the zeros we found to the answer options provided and have determined that the correct answer is B. $2\sqrt{2}, \sqrt{2}$.

Q: What is a polynomial?

A: A polynomial is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. It is a mathematical expression that can be written in the form $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants.

Q: What are the zeros of a polynomial?

A: The zeros of a polynomial are the values of the variable that make the polynomial equal to zero. In other words, they are the solutions to the equation $P(x) = 0$.

Q: How do I find the zeros of a polynomial?

A: There are several ways to find the zeros of a polynomial, including factoring, using the quadratic formula, and graphing. Factoring involves finding two numbers whose product is the constant term and whose sum is the coefficient of the linear term. The quadratic formula involves using the formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ to find the solutions.

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula that is used to find the solutions to a quadratic equation. It is given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where $a$, $b$, and $c$ are the coefficients of the quadratic equation.

Q: How do I use the quadratic formula?

A: To use the quadratic formula, you need to plug in the values of $a$, $b$, and $c$ into the formula. Then, simplify the expression and solve for $x$.

Q: What are the steps to find the zeros of a polynomial?

A: The steps to find the zeros of a polynomial are:

1. Set the polynomial equal to zero: $P(x) = 0$.
2. Factor the polynomial, if possible.
3. Use the quadratic formula to find the solutions.
4. Simplify the solutions and solve for $x$.

Q: Can I use a calculator to find the zeros of a polynomial?

A: Yes, you can use a calculator to find the zeros of a polynomial. Most calculators have a built-in function to solve quadratic equations.

Q: What are some common mistakes to avoid when finding the zeros of a polynomial?

A: Some common mistakes to avoid when finding the zeros of a polynomial include:

• Not setting the polynomial equal to zero.
• Not factoring the polynomial, if possible.
• Not using the quadratic formula correctly.
• Not simplifying the solutions.

Q: Can I find the zeros of a polynomial with a negative leading coefficient?

A: Yes, you can find the zeros of a polynomial with a negative leading coefficient. The process is the same as finding the zeros of a polynomial with a positive leading coefficient.

Q: Can I find the zeros of a polynomial with a complex coefficient?

A: Yes, you can find the zeros of a polynomial with a complex coefficient. The process is the same as finding the zeros of a polynomial with a real coefficient.

Q: What are some real-world applications of finding the zeros of a polynomial?

A: Some real-world applications of finding the zeros of a polynomial include:

• Modeling population growth and decline.
• Modeling the motion of objects.
• Finding the maximum or minimum value of a function.

Q: Can I find the zeros of a polynomial with a rational coefficient?

A: Yes, you can find the zeros of a polynomial with a rational coefficient. The process is the same as finding the zeros of a polynomial with a real coefficient.

Q: Can I find the zeros of a polynomial with a polynomial coefficient?

A: Yes, you can find the zeros of a polynomial with a polynomial coefficient. The process is the same as finding the zeros of a polynomial with a real coefficient.

Conclusion

In conclusion, finding the zeros of a polynomial is an important concept in algebra. It has many real-world applications and can be used to model a wide range of phenomena. By following the steps outlined in this article, you can find the zeros of a polynomial with ease.