Using the quadratic formula always gives you two solutions, because the plus/minus sign means you’re both adding and subtracting and getting two completely different answers. When the number underneath the square-root sign in the quadratic formula is negative, the answers are called complex conjugates. One is r + si and the other is r – si. These numbers have both real (the r) and imaginary (the si) parts.
The complex number system consists of all numbers r + si where r and s are real numbers. Observe that when s = 0, you simply have the real numbers. Therefore the real numbers are a subset of the complex number system. The fundamental theorem of algebra says that every polynomial function has at least one root in the complex number system.
The highest degree of a polynomial gives you the highest possible number of distinct complex roots for the polynomial. Between this fact and Descartes’s rule of signs, you can get an idea of how many imaginary roots a polynomial has.Here’s how Descartes’s rule of signs can give you the numbers of possible real roots, both positive and negative:
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Positive real roots. For the number of positive real roots, look at the polynomial, written in descending order, and count how many times the sign changes from term to term. This value represents the maximum number of positive roots in the polynomial. For example, in the polynomial f(x) = 2x4 – 9x3 – 21x2 + 88x + 48, you see two changes in sign (don’t forget to include the sign of the first term!) — from the first term (+2x4) to the second (-9x3) and from the third term (-21x2) to the fourth term (88x). That means this equation can have up to two positive solutions.
Descartes’s rule of signs says the number of positive roots is equal to changes in sign of f(x), or is less than that by an even number (so you keep subtracting 2 until you get either 1 or 0). Therefore, the previous f(x) may have 2 or 0 positive roots.
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Negative real roots. For the number of negative real roots, find f(–x) and count again. Because negative numbers raised to even powers are positive and negative numbers raised to odd powers are negative, this change affects only terms with odd powers. This step is the same as changing each term with an odd degree to its opposite sign and counting the sign changes again, which gives you the maximum number of negative roots. The example equation becomes f(–x) = 2x4 + 9x3 – 21x2 – 88x + 48, which changes signs twice. There can be, at most, two negative roots. However, similar to the rule for positive roots, the number of negative roots is equal to the changes in sign for f(–x), or must be less than that by an even number. Therefore, this example can have either 2 or 0 negative roots.
For example, the polynomial f(x) = 2x4 – 9x3 – 21x2 + 88x + 48 has a degree of 4, with two or zero positive real roots, and two or zero negative real roots. With this information, you can pair up the possible situations:
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Two positive and two negative real roots, with zero imaginary roots
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Two positive and zero negative real roots, with two imaginary roots
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Zero positive and two negative real roots, with two imaginary roots
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Zero positive and zero negative real roots, with four imaginary roots
| Positive real roots | Negative real roots | Imaginary roots |
|---|---|---|
| 2 | 2 | 0 |
| 2 | 0 | 2 |
| 0 | 2 | 2 |
| 0 | 0 | 4 |
The fundamental theorem of algebra gives the total number of complex roots (say there are seven); Descartes’s rule of signs tells you how many possible real roots exist and how many of them are positive and negative (say there are, at most, two positive roots but only one negative root). Now, assume you’ve found them all: x = 1, x = 7, and x = –2. These roots are real, but they’re also complex because they can all be rewritten.
The first two columns in the chart find the real roots and classify them as positive or negative. The third column is actually finding, specifically, the non-real numbers: complex numbers with non-zero imaginary parts.