Objectives Determine roots of polynomial equations. Apply the Fundamental Theorem of Algebra.
Polynomial in One Variable a 0, a 1, a 2,... a n : complex numbers (real or imaginary) a 0 ≠0 n: non-negative integer
Definitions The degree of a polynomial is the greatest exponent of the variable. The leading coefficient is the coefficient of the variable with the greatest exponent. If all the coefficients are real numbers, it is a polynomial function. The values of x where f(x)=0 are called the zeros (x-intercepts).
What is the degree? 4 What is the leading coefficient? 3 Is -2 a zero of f(x)? no Example
Imaginary Numbers Complex numbers: a+bi (a and b are real numbers) Pure imaginary number: a=0 and b≠0
Fundamental Theorem of Algebra Every polynomial equation with degree>0 has at least one root in the set of complex numbers Corollary: A polynomial of degree n has exactly n complex roots.
Maximum Number of Roots Degree: 1Degree: 2 Degree: 3Degree: 4Degree: 5
Determining Roots You can’t determine imaginary roots from the graph – you can only see the real roots. Imaginary roots come in pairs. An equation with odd degree must have a real root.
Finding the Polynomial If you know the roots, you can find the polynomial. (x-a)(x-b)=0
Example Write a polynomial equation of least degree with roots 2, 3i and -3i. (x-2)(x-3i)(x+3i)=0 (x-2)(x 2 -9i 2 )=0 (x-2)(x 2 +9)=0 x 3 -2x 2 +9x-18=0 Does the equation have odd or even degree? Odd How many times does the graph cross the x-axis? Once
Example State the number of complex roots of the equation 32x 3 - 32x 2 + 4x - 4=0. Find the roots and graph. 32x 3 - 32x 2 + 4x - 4=0 32x 2 (x-1)+4(x-1)=0 (32x 2 +4)(x-1)=0 x 2 =-4/32 or x=1 x=±√-1/8 x=±i/2√2 =±i√2/4 Roots are 1 and ±i√2/4
Meterorology Example (#5) A meteorologist sends a temperature probe on a small weather rocket through a cloud layer. The launch pad for the rocket is 2 feet off the ground. The height of the rocket after launching is modeled by the equation h=-16t 2 +232t+2 where h is the height of the rocket in feet and t is the elapsed time in seconds. When will the rocket be 114 feet above the ground?
Solution Find t when h=114. h=-16t 2 +232t+2 114=-16t 2 +232t+2 0=-16t 2 +232t-112 0=-8(2t 2 -29t+14) 0=-8(2t-1)(t-14) 2t-1=0 or t-14=0 t=1/2 or t=14 (xscl=5, yscl=100)