How many complex and real zeros

We are now in the position to define the complex numbers. The arithmetic of complex numbers is as you would expect. The only things you need to remember are the two properties in Definition 3. The next example should help recall how these animals behave. Perform the indicated operations. A couple of remarks about the last example are in order. The properties of the conjugate are summarized in the following theorem.

Essentially, Theorem 3. Next, we compute the left and right hand sides of each equation and check to see that they are the same. The proof of the first property is a very quick exercise. The proof for multiplication works similarly. The proof that the conjugate works well with powers can be viewed as a repeated application of the product rule, and is best proved using a technique called Mathematical Induction.

The last property is a characterization of real numbers. We now return to the business of zeros. Two things are important to note. Do you see why? Next, we note that in Example 3.

This demonstrates that the factor theorem holds even for non-real zeros, i. It turns out that polynomial division works the same way for all complex numbers, real and non-real alike, so the Factor and Remainder Theorems hold as well. But how do we know if a general polynomial has any complex zeros at all? We have many examples of polynomials with no real zeros. Can there be polynomials with no zeros whatsoever? The answer to that last question is ''No. The Fundamental Theorem of Algebra is an example of an 'existence' theorem in Mathematics.

Like the Intermediate Value Theorem, Theorem 3. Since the total number of zeros of f x is 4 , that means it has 0 or 2 non-Real Complex zeros. How do you determine the number of complex roots of a polynomial of degree n? George C. Sep 20, See explanation Related questions How do you solve radical equations? What are Radical Equations? How do you solve radical equations with cube roots? So, the roots are.

The polynomial has a real zero at 1. Find the other two zeros. If this polynomial has a real zero at 1. We can figure out what this is this way:. To finish solving, we can use the quadratic formula with the resulting quadratic, :. If the real zero of the polynomial is 3, what are the complex zeros?

We know that the real zero of this polynomial is 3, so one of the factors must be. To find the other factors, we can divide the original polynomial by , either by long division or synthetic division:. This gives us a second factor of which we can solve using the quadratic formula:.

The polynomial intersects the x-axis at point. Find the other two solutions. Since we know that one of the zeros of this polynomial is 3, we know that one of the factors is. To find the other two zeros, we can divide the original polynomial by , either with long division or with synthetic division:.

This gives us the second factor of. We can get our solutions by using the quadratic formula:. Find all the real and complex zeroes of the following equation:. First, factorize the equation using grouping of common terms:. Next, setting each expression in parenthesis equal to zero yields the answers. Find all the zeroes of the following equation and their multiplicity:. First, pull out the common t and then factorize using quadratic factoring rules:. This equation has solutions at two values: when and when.

Therefore, But since the degree on the former equation is one and the degree on the latter equation is two, the multiplicities are 1 and 2 respectively.

Find a fourth degree polynomial whose zeroes are -2, 5, and. This one is a bit of a journey. The expressions for the first two zeroes are easily calculated, and respectively. The last expression must be broken up into two equations:. Finally, we multiply together all of the parenthesized expressions, which multiplies out to. The third degree polynomial expression has a real zero at. Find all of the complex zeroes. First, factor the expression by grouping:. To find the complex zeroes, set the term equal to zero:.