Polynomials are algebraic expressions that consist of terms with variables raised to positive integer powers, such as x2 + 3x − 5. The degree of a polynomial is the highest power of the variable that appears in the polynomial. For example, the degree of x2 + 3x − 5 is 2, because the highest power of x is 2.
Complex numbers are numbers that have both a real part and an imaginary part, such as 2 + 3i, where i is the square root of −1. Complex numbers can be used to represent points on a plane, where the real part is the horizontal coordinate and the imaginary part is the vertical coordinate.
One of the most important results in algebra is the Fundamental Theorem of Algebra, which states that every polynomial of degree n with complex coefficients has exactly n roots in the complex numbers, where a root is a value that makes the polynomial equal to zero. For example, x2 − 9 has two roots: 3 and −3, because (3)2 − 9 = 0 and (−3)2 − 9 = 0.
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Why Complex Numbers Are Needed to Find All Roots
Some polynomials have roots that are not real numbers, but complex numbers. For example, x2 + 1 has no real roots, because there is no real number x that satisfies x2 + 1 = 0. However, it has two complex roots: i and −i, because (i)2 + 1 = 0 and (−i)2 + 1 = 0.
If we only consider real numbers, we might think that some polynomials have fewer roots than their degree. For example, x3 − x has degree 3, but it seems to have only two real roots: 1 and −1, because (1)3 − (1) = 0 and (−1)3 − (−1) = 0. However, if we use complex numbers, we can find another root: 0, because (0)3 − (0) = 0. Therefore, x3 − x has three roots in total, as expected by the Fundamental Theorem of Algebra.
How to Find Complex Roots Using Quadratic Formula
One way to find complex roots of a polynomial is to use the quadratic formula, which applies to any polynomial of degree 2 with real or complex coefficients. The quadratic formula states that if ax2 + bx + c = 0, where a ≠ 0, then the roots are given by:
�=−�±�2−4��2�x=2a−b±b2−4ac
The expression under the square root sign, b2 −4ac, is called the discriminant. It determines whether the roots are real or complex. If the discriminant is positive, then there are two distinct real roots. If the discriminant is zero, then there is one repeated real root. If the discriminant is negative, then there are two distinct complex roots.
For example, consider x2 −x+1 = 0. The coefficients are a = 1, b = −1, and c = 1. The discriminant is b2 −4ac = (−1)2 −4(1)(1) = −3. Since the discriminant is negative, there are two complex roots. Using the quadratic formula, we get:
�=−(−1)±(−1)2−4(1)(1)2(1)x=2(1)−(−1)±(−1)2−4(1)(1)
�=1±−32x=21±−3
�=1±�32x=21±i3
Therefore, the two complex roots are 1+�3221+i3 and 1−�3221−i3.
How Complex Roots Come in Pairs
Another interesting property of complex roots is that they always come in pairs that are conjugates of each other. A conjugate of a complex number is obtained by changing the sign of the imaginary part. For example, the conjugate of 2 + 3i is 2 − 3i.
The reason why complex roots come in pairs is that if z is a root of a polynomial with real coefficients, then so is its conjugate �‾z. This can be shown by using the fact that �‾�=��‾zn=zn for any positive integer n and any complex number z. For example,
(�)5‾=�5‾=−�‾=�(i)5=i5=−i=i
If z is a root of a polynomial with real coefficients, then we can write the polynomial as:
����+��−1��−1+…+�1�+�0=0anzn+an−1zn−1+…+a1z+a0=0
where an, an−1, …, a1, a0 are real numbers. Taking the conjugate of both sides, we get:
����+��−1��−1+…+�1�+�0‾=0‾anzn+an−1zn−1+…+a1z+a0=0
Using the properties of conjugates, we can simplify this as:
��‾��‾+��−1‾��−1‾+…+�1‾�‾+�0‾=0anzn+an−1zn−1+…+a1z+a0=0
Since the coefficients are real, their conjugates are equal to themselves. Therefore, we have:
����‾+��−1��−1‾+…+�1�‾+�0=0anzn+an−1zn−1+…+a1z+a0=0
This shows that �‾z is also a root of the polynomial.
For example, consider x2 −x+1 = 0 again. We found that one of the roots is 1+�3221+i3. The conjugate of this root is 1−�3221−i3, which is also a root of the polynomial. You can check this by plugging it into the polynomial and simplifying:
(1−�32)2−(1−�32)+1=0(21−i3)2−(21−i3)+1=0
(1−�3)24−(1−�3)2+1=04(1−i3)2−2(1−i3)+1=0
(1−2�3−3)4−(2−2�3)4+1=04(1−2i3−3)−4(2−2i3)+1=0
−4�34+1=04−4i3+1=0
−�3+1=0−i3+1=0
Conclusion
In this article, we have seen how the degree of a polynomial determines the number of complex roots it has, according to the Fundamental Theorem of Algebra. We have also seen how to find complex roots using the quadratic formula, and how complex roots come in pairs that are conjugates of each other. Complex numbers are useful tools for solving polynomial equations that have no real solutions. They also help us understand the behavior and shape of polynomial functions in the complex plane.
