Introduction to Fourth Degree Polynomials

Polynomials are algebraic expressions consisting of variables and coefficients. When dealing with polynomials of degree four, or fourth degree polynomials, it can seem like a daunting task to find the roots. However, with specific techniques, you can simplify the process even when certain terms are missing. This article will guide you through the method of solving a fourth degree polynomial without the third degree and first degree terms, using a simple substitution technique.

Identifying the Polynomial Form

A fourth degree polynomial, also known as a quartic polynomial, can be generally expressed in the form:

ax^4 bx^2 c

In this form, you might notice that the polynomial does not contain the x^3 and the x (first degree) terms. This absence can make the polynomial appear more complex, but fortunately, it can be approached with ease using a strategic substitution technique.

The Quadratic Substitution Technique

To solve a fourth degree polynomial with the form ax^4 bx^2 c, we can use a substitution method. This method simplifies the polynomial to a quadratic form, which is easier to solve. Here's how it works:

Step 1: Define the substitution

Substitute y x^2. This transforms the polynomial from a fourth degree equation to a second degree equation (quadratic equation):

ay^2 by c

Step 2: Solve the quadratic equation

Once the polynomial has been transformed into a quadratic form, you can solve it using standard quadratic equation solving methods. The quadratic formula is:

y [-b ± sqrt(b^2 - 4ac)] / (2a)

Here, a, b, and c are the coefficients of the quadratic equation ay^2 by c. Substitute these values into the formula to find the values of y, which are the solutions to the original equation in terms of y.

Step 3: Return to the original variable

Once you have the values of y, substitute back x^2 y to find the values of x. This step involves taking the square root of both sides, giving you both positive and negative roots:

x ±sqrt(y)

Thus, by solving for y and then for x, you can obtain the roots of the original fourth degree polynomial.

Examples and Applications

Let's walk through an example to better understand the process:

Example: Solve the polynomial 6x^4 - 5x^2 - 4 0

Step 1: Substitute y x^2

6y^2 - 5y - 4 0

Step 2: Solve the quadratic equation

Here, a 6, b -5, and c -4. Substitute these values into the quadratic formula:

y [5 ± sqrt(25 96)] / 12 [5 ± sqrt(121)] / 12 (5 ± 11) / 12

This gives us two values for y

y1 (5 11) / 12 16 / 12 4 / 3

y2 (5 - 11) / 12 -6 / 12 -1 / 2

Step 3: Convert back to x

For y1 4 / 3 and y2 -1 / 2, we have:

x1 ±sqrt(4/3) ±(2sqrt(3))/3

x2 ±sqrt(-1/2) (no real solutions for negative y)

Thus, the real roots of the polynomial are:

x ±(2sqrt(3))/3

Understanding and applying this technique can significantly simplify the process of solving higher degree polynomials, especially when certain terms are missing. This method is a powerful tool in algebra and can be applied in various fields, including physics, engineering, and computer science.

Conclusion

Solving fourth degree polynomials without third and first degree terms can be made straightforward with the quadratic substitution technique. By substituting x^2 y, you can transform the polynomial into a quadratic equation and solve for y, then convert back to x. This method not only simplifies the problem but also provides a clear and systematic approach. Applying this technique can make complex equations more manageable and enhance your problem-solving skills in mathematics and its applications.