# Write an equation for a fourth degree polynomial with no real zeros

We can infer that the numerators of the rational roots will always be factors of the constant term and the denominators will be factors of the leading coefficient.

### Quartic function with no real zeros

In a synthetic division table do the multiplications in our head and drop the middle row just writing down the third row and since we will be going through the process multiple times we put all the rows into a table. Notice that two of the factors of the constant term, 6, are the two numerators from the original rational roots: 2 and 3. In optics, Alhazen's problem is "Given a light source and a spherical mirror, find the point on the mirror where the light will be reflected to the eye of an observer. The Fundamental Theorem of Algebra tells us that every polynomial function has at least one complex zero. Also note the presence of the two turning points. The same is true for the intersection of a line and a torus. Finding the Zeros of Polynomial Functions The Rational Zero Theorem helps us to narrow down the list of possible rational zeros for a polynomial function. Here then is a list of all possible rational zeroes of this polynomial. There are two sign changes, so there are either 2 or 0 positive real roots. So, the first thing to do is actually to list all possible factors of 1 and 6. Use the Rational Zero Theorem to list all possible rational zeros of the function.

Evaluating a Polynomial Using the Remainder Theorem In the last section, we learned how to divide polynomials. Notice that two of the factors of the constant term, 6, are the two numerators from the original rational roots: 2 and 3.

Use synthetic division to evaluate a given possible zero by synthetically dividing the candidate into the polynomial.

Recall that the Division Algorithm.

### How to find a 4th degree polynomial

This means that, since there is a 3rd degree polynomial, we are looking at the maximum number of turning points. At this point we can solve this directly for the remaining zeroes. It tells us how the zeros of a polynomial are related to the factors. In this section, we will discuss a variety of tools for writing polynomial functions and solving polynomial equations. Confirm that the remainder is 0. A complex number is not necessarily imaginary. Find the zeros of the quadratic function. Once the polynomial has been completely factored, we can easily determine the zeros of the polynomial.

We will use synthetic division to evaluate each possible zero until we find one that gives a remainder of 0. Use the factors to determine the zeros of the polynomial.

Figure 1 Use the Fundamental Theorem of Algebra Now that we can find rational zeros for a polynomial function, we will look at a theorem that discusses the number of complex zeros of a polynomial function.

This is actually easier than it might at first appear to be.

## 4th degree polynomial equation

Use the Rational Zero Theorem to find rational zeros. Solution: Begin by determining the number of sign changes. We can now use polynomial division to evaluate polynomials using the Remainder Theorem. To do the evaluations we will build a synthetic division table. This pair of implications is the Factor Theorem. Once the polynomial has been completely factored, we can easily determine the zeros of the polynomial. Example 1 Verify that the roots of the following polynomial satisfy the rational root theorem. In a synthetic division table do the multiplications in our head and drop the middle row just writing down the third row and since we will be going through the process multiple times we put all the rows into a table. What this fact is telling us is that if we evaluate the polynomial at two points and one of the evaluations gives a positive value i. So we can shorten our list. The quadratic is a perfect square. Analysis of the Solution Look at the graph of the function f. By the Factor Theorem, these zeros have factors associated with them.

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