Finding Zeros Of Polynomial Functions In Exact Form

Finding the zeros of a polynomial function is a fundamental problem in algebra with applications across various fields, including engineering, physics, and computer science. In this article, we will delve into the process of finding the zeros of a polynomial function, specifically focusing on the equation $t(x) = x^4 - 10x^2 - 56$. We will explore the techniques used to solve such equations and express the solutions in exact form.

The primary goal here is to determine the values of $x$ for which the function $t(x)$ equals zero. These values are also known as the roots or solutions of the polynomial equation. For polynomial functions of higher degrees, such as the fourth-degree polynomial in this case, finding the zeros can be more intricate than solving quadratic equations. However, by employing suitable algebraic manipulations and techniques, we can arrive at the solutions.

Let's embark on the journey of unraveling the zeros of the given polynomial function, ensuring that we provide the answers in their exact form, separated by commas if there are multiple solutions, and acknowledging the possibility of no applicable solutions if the situation warrants it.

Solving $t(x) = x^4 - 10x^2 - 56$

To find the zeros of the polynomial function $t(x) = x^4 - 10x^2 - 56$, we need to solve the equation $x^4 - 10x^2 - 56 = 0$. This is a quartic equation, but it has a special form that allows us to solve it as a quadratic equation by using a substitution. Let's set $y = x^2$. Then, the equation becomes:

$y^2 - 10y - 56 = 0$

This is a quadratic equation in $y$, which we can solve using factoring, completing the square, or the quadratic formula. Let's try factoring first. We are looking for two numbers that multiply to -56 and add to -10. These numbers are -14 and 4. So, we can factor the quadratic equation as follows:

$(y - 14)(y + 4) = 0$

This gives us two possible values for $y$:

$y = 14$ or $y = -4$

Now, we need to substitute back $x^2$ for $y$ and solve for $x$:

Case 1: $y = 14$

$x^2 = 14$

Taking the square root of both sides, we get:

$x = \pm\sqrt{14}$

So, we have two real solutions: $x = \sqrt{14}$ and $x = -\sqrt{14}$.

Case 2: $y = -4$

$x^2 = -4$

Taking the square root of both sides, we get:

$x = \pm\sqrt{-4}$

Since the square root of a negative number is imaginary, we have two complex solutions:

$x = \pm 2i$

Where $i$ is the imaginary unit, defined as $i = \sqrt{-1}$.

Therefore, the zeros of the polynomial function $t(x) = x^4 - 10x^2 - 56$ are $\sqrt{14}$, $-\sqrt{14}$, $2i$, and $-2i$.

Expressing the Zeros in Exact Form

As we found in the previous section, the zeros of the polynomial function $t(x) = x^4 - 10x^2 - 56$ are $\sqrt{14}$, $-\sqrt{14}$, $2i$, and $-2i$. These values are already in their exact form, meaning they are not approximated decimals but rather precise representations using radicals and imaginary units where necessary.

• $\sqrt{14}$ is the square root of 14, which is an irrational number and cannot be expressed as a simple fraction. Thus, leaving it in this form is the exact representation.
• Similarly, $-\sqrt{14}$ is the negative square root of 14, also an irrational number in its exact form.
• $2i$ and $-2i$ are imaginary numbers, where $i$ represents the imaginary unit. These are also expressed in their exact forms.

When providing the answer, we list these zeros separated by commas:

$\sqrt{14}$, $-\sqrt{14}$, $2i$, $-2i$

This ensures that we are communicating the complete set of zeros for the given polynomial function in a clear and precise manner.

Importance of Exact Form

Expressing the zeros of a polynomial in exact form is crucial in mathematics for several reasons. Firstly, exact forms provide precise solutions without any rounding errors. Approximations, while useful in some contexts, can lead to inaccuracies, especially in complex calculations where errors can accumulate. Secondly, exact forms allow for a deeper understanding of the nature of the roots. For instance, recognizing a root as $\sqrt{2}$ rather than 1.414 helps in understanding its algebraic properties. Lastly, in many mathematical contexts, particularly in algebra and calculus, exact forms are necessary for further manipulations and proofs.

Conclusion

In this article, we have demonstrated how to find the zeros of the polynomial function $t(x) = x^4 - 10x^2 - 56$ and express them in exact form. By using the substitution method to transform the quartic equation into a quadratic equation, we were able to find the four zeros: $\sqrt{14}$, $-\sqrt{14}$, $2i$, and $-2i$. These zeros represent the values of $x$ for which the function $t(x)$ equals zero. Expressing these solutions in exact form is essential for maintaining precision and facilitating further mathematical analysis. The process outlined here can be applied to other similar polynomial equations, providing a valuable tool for solving algebraic problems.