Definitions
an expression that involves the operations of addition, subtraction, and multiplication
What is a polynomial?
the highest power of the variable in a polynomial of one variable
What is the degree of a polynomial?
every one-variable polynomial has at least one complex root (remember that every real number is also a complex number), and furthermore every polynomial of degree n has n complex roots
What is the Fundamental Theorem of Algebra?
How many solutions does the equation 3x=x2-4x+6 have?
What is 2?
What is the conjugate of -3+2i?
-3-2i
the value of b2-4ac
What is the discriminant?
the square root of -1
What is the imaginary number i?
If x=a is a zero of the polynomial p(x), then (x-a) is a factor of the polynomial, and if (x-a) is a factor of the polynomial, then x=a is a zero of the polynomial
What is the Factor Theorem?
Solve 2x2-2x+5=0
What is x=1/2+3/2i, 1/2-3/2i ?
Factor x2+1
(x+i)(x-i)
solutions of the equation p(x)=0
What is the root (zeroes) of a polynomial function?
the set of numbers that solve equations of the form x2=(a negative number)
What is imaginary numbers?
For any number c, when a polynomial p(x) is divided by (x-c), the remainder is p(c).
What is the Remainder Theorem?
What is the absolute value of 4-3i?
5
Sketch a fifth-degree polynomial with four complex roots and one real root.
[Answers vary]
f(x)=(an)xn+(an-1)x(n-1)+...+(a1)x1+a0
What is the general equation of an n-degree polynomial?
the sum of a real number (other than zero) and an imaginary number, this sum can be written in the form a+bi
What is complex numbers?
For any polynomial with integral coefficients, if an integer is a zero of the polynomial, it must be a factor of the constant term
What is the Integral Zero Theorem?
Sketch the graph of the function f(x) = (x + 1)(x – 3)2(x + 4)3. Describe the function as completely as you can.
What is a single root at (-1,0), double root at (3,0), triple root at (-4,0)
If you are shown the graph of a polynomial, how can you tell if the roots are complex or real? Explain.
Existence of x-intercept (where the graph cross the x-axis) means that the roots are real.
a-bi and a+bi, the sum and product of these are always real numbers
What is complex conjugates?
a graph where real numbers are on the horizontal axis and imaginary numbers are on the vertical axis
What is the complex plane?
Free 500 points
Yay?
Simplify i1029
-i
x=3 is a root of f(x)=x3-4x2-17x+60. Factor the polynomial into three linear factors.
(x-3)(x-5)(x+4)