Lesson 3.6 Complex Roots of Polynomials

1. Lesson 3.6 Complex Roots of Polynomials At the conclusion of this lesson you will be able to: • Write complex numbers in the form of a + bi • Find zeros of quadratic functions involving complex numbers • Determine how many complex roots a polynomial may have before • factoring it completely • Write polynomial functions that have complex roots.

2. We have seen several times where a graph does not intersect the x-axis. This means that there is no real number that makes a function equal to 0. Ex: To examine the zeros of a function like this we must extend our number system once more to the set of Complex Numbers. Before we can define complex numbers, we must introduce the number . From this we get cannot be a real number because there is no real number that satisfies

3. Complex Numbers The set of complex numbers consists of all expressions of the form Where a and b are real numbers. This set is often denoted C. The number a is called the real part of the complex number z, and the number b is called the imaginary part of z. When b = 0, the complex number is a real number, therefore, all real numbers are complex numbers. Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal. a + bi = c + di if and only if a = c and b = d

4. Addition, Subtraction and Multiplication of Complex Numbers Addition and subtraction is done just like other numbers: combine like terms. Multiplication is just like multiplying binomials: FOIL method. Ex 1: Determine the sum and product of the complex numbers: Sum: Difference: Product:

5. Recall: quadratic formula: We use the quadratic formula to find the zeros of a quadratic function. When there are two real solutions. When there is one real solution. When there are two complex solutions. How to recover the complex roots from the quadratic formula: Multiplied both sides by -1 This is now a real number

6. Ex 2: Determine all the solutions to the equation Solution: a = 1 b = -4 c = 13 Recall:turning a zero into a factor: If x = 3, the factor would be… (x – 3). Now, turn the complex roots into factors. NOTE: The two complex solutions have the same real part and the imaginary parts only differ by sign. These are called Complex Conjugates.

7. Complex Conjugates The complex conjugate of the complex number is the complex number For any complex number z, which are both real numbers.

8. Ex 3: Determine the complex conjugates of and and then find: Solution: Conjugate Results For any pair of complex numbers, z and w, and for any integer n, we have

9. Complex roots come in conjugate pairs. Ex: If x = 2 – 3i is a complex root of a polynomial, then so is x = 2 + 3i. Ex 4: One solution to the equation is the complex number . Determine all the solutions to this equation. Solution: If x = 1 + 2i is a solution then so is x = 1 – 2i. Now, turn them into factors and simplify. (x – (1 + 2i)) = (x – 1 – 2i) (x – (1 – 2i)) = (x – 1 + 2i) The product of these two new factors is the divisor (D(x)) for your division algorithm. (x – 1 – 2i)(x – 1 + 2i) = x2 – 2x + 5 = D(x) Notice that when complex conjugates are multiplied that the i’s cancel out.

10. Now, divide using the division algorithm. Factor Q(x) a = 1 b = -4 c = 1 We can see that we need to use the quadratic formula What if the directions asked to factor completely? turn solutions into factors.

11. Ex 5: Factor completely. Solution: We need to find at least one zero on our own before we can do the division algorithm. (RATIONAL ZERO TEST) x = 3, the factor is: (x - 3) which is your D(x) for your division algorithm. Now use the quad. formula to find other zeros.