Zeroes 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 Are:A. 2 , 2 2, \sqrt{2} 2 , 2 ​ 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

Solving the Polynomial: Finding the Zeroes of $p(x) = x^2 - 3 \sqrt{2} x + 4$

In mathematics, a polynomial is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. The zeroes 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 zeroes of the given polynomial $p(x) = x^2 - 3 \sqrt{2} x + 4$.

Understanding the Quadratic Formula

To find the zeroes of a quadratic polynomial, we can use the quadratic formula. The quadratic formula is given by:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

where $a$, $b$, and $c$ are the coefficients of the quadratic polynomial. In this case, the polynomial is $p(x) = x^2 - 3 \sqrt{2} x + 4$, so we have:

$a = 1$, $b = -3 \sqrt{2}$, and $c = 4$

Applying the Quadratic Formula

Now that we have the values of $a$, $b$, and $c$, we can plug them into the quadratic formula to find the zeroes of the polynomial.

$x = \frac{-(-3 \sqrt{2}) \pm \sqrt{(-3 \sqrt{2})^2 - 4(1)(4)}}{2(1)}$

Simplifying the expression, we get:

$x = \frac{3 \sqrt{2} \pm \sqrt{18 - 16}}{2}$

$x = \frac{3 \sqrt{2} \pm \sqrt{2}}{2}$

Finding the Zeroes

Now that we have the expression for the zeroes, we can simplify it further to find the actual values.

$x = \frac{3 \sqrt{2} + \sqrt{2}}{2}$ or $x = \frac{3 \sqrt{2} - \sqrt{2}}{2}$

Simplifying the expressions, we get:

$x = \frac{4 \sqrt{2}}{2}$ or $x = \frac{2 \sqrt{2}}{2}$

$x = 2 \sqrt{2}$ or $x = \sqrt{2}$

Conclusion

In conclusion, the zeroes of the polynomial $p(x) = x^2 - 3 \sqrt{2} x + 4$ are $2 \sqrt{2}$ and $\sqrt{2}$. These values satisfy the equation $p(x) = 0$, making them the zeroes of the polynomial.

Answer

The correct answer is B. $2 \sqrt{2}, \sqrt{2}$
Frequently Asked Questions: Zeroes of the Polynomial $p(x) = x^2 - 3 \sqrt{2} x + 4$

In the previous article, we discussed how to find the zeroes of the polynomial $p(x) = x^2 - 3 \sqrt{2} x + 4$. In this article, we will answer some frequently asked questions related to the topic.

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula that is used to find the zeroes of a quadratic polynomial. It is given by:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

where $a$, $b$, and $c$ are the coefficients of the quadratic polynomial.

Q: How do I apply the quadratic formula to find the zeroes of a polynomial?

A: To apply the quadratic formula, you need to plug in the values of $a$, $b$, and $c$ into the formula. Then, simplify the expression to find the actual values of the zeroes.

Q: What are the coefficients of the polynomial $p(x) = x^2 - 3 \sqrt{2} x + 4$?

A: The coefficients of the polynomial $p(x) = x^2 - 3 \sqrt{2} x + 4$ are:

$a = 1$, $b = -3 \sqrt{2}$, and $c = 4$

Q: How do I simplify the expression for the zeroes of the polynomial?

A: To simplify the expression for the zeroes, you need to combine like terms and simplify the square root expression.

Q: What are the zeroes of the polynomial $p(x) = x^2 - 3 \sqrt{2} x + 4$?

A: The zeroes of the polynomial $p(x) = x^2 - 3 \sqrt{2} x + 4$ are $2 \sqrt{2}$ and $\sqrt{2}$.

Q: How do I check if a value is a zero of a polynomial?

A: To check if a value is a zero of a polynomial, you need to plug the value into the polynomial and check if the result is equal to zero.

Q: What is the significance of the zeroes of a polynomial?

A: The zeroes of a polynomial are the values of the variable that make the polynomial equal to zero. They are important in many areas of mathematics and science, including algebra, calculus, and physics.

Q: Can I use the quadratic formula to find the zeroes of any polynomial?

A: Yes, you can use the quadratic formula to find the zeroes of any quadratic polynomial. However, if the polynomial is not quadratic, you may need to use other methods to find its zeroes.

Q: Are the zeroes of a polynomial always real numbers?

A: No, the zeroes of a polynomial are not always real numbers. In some cases, the zeroes may be complex numbers.

Conclusion

In conclusion, the quadratic formula is a powerful tool for finding the zeroes of a polynomial. By understanding how to apply the quadratic formula and simplify the expression for the zeroes, you can solve many problems in algebra and other areas of mathematics.