4-1 Polynomial Functions

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Presentation transcript:

4-1 Polynomial Functions

Objectives Determine roots of polynomial equations. Apply the Fundamental Theorem of Algebra.

Polynomial in One Variable a0, a1, a2, . . . an: complex numbers (real or imaginary) a0≠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).

Example What is the degree? 4 What is the leading coefficient? 3 Is -2 a zero of f(x)? no

Imaginary Numbers Complex numbers: a+bi (a and b are real numbers) Pure imaginary number: a=0 and b≠0

Imaginary Numbers

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: 1 Degree: 2 Degree: 3 Degree: 4 Degree: 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)(x2-9i 2)=0 (x-2)(x2+9)=0 x3-2x2+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 32x3 - 32x2 + 4x - 4=0. Find the roots and graph. 32x3 - 32x2 + 4x - 4=0 32x2(x-1)+4(x-1)=0 (32x2+4)(x-1)=0 x2=-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=-16t2+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=-16t2+232t+2 114=-16t2+232t+2 2t-1=0 or t-14=0 t=1/2 or t=14 (xscl=5, yscl=100)

Homework page 210 #15-47 odds Don’t graph 39-47