6.7
Using the Fundamental Theorem of Algebra
When all real and imaginary solutions are counted, a polynomial of degree n
has exactly n solutions when you count both the real and imaginary solutions.
Quadratic Equations:
Degree =_____________
Find ALL zeroes of the equations below:
1.) y = x 2 − 4
2.) y = x 2 + 4
Solutions:________
3.)
Solutions:___________
y = x2 − 2x + 1
Solutions: _____________
Special Case Called_________
Quadratic Equations will always have a total of ________ real and imaginary solutions.
Cubic Equations:
Degree = _______________
Find all zeroes of the equations below:
1.) y = x 3 − x 2 − 2 x
Solutions: ______________
2.)
y = x 3 + 3 x 2 + 16 x + 48
Solutions:______________
3.)
y = x 3 + x 2 − x + 15
Solutions: ____________
6.7
Using the Fundamental Theorem of Algebra
Cubic Equations will always have a total of ________ real and imaginary solutions.
Quartic Equations
Degree=___________
Find all zeroes of the equations below:
1.) y = x 4 + 5 x 3 + 5 x 2 − 5 x − 6
Solutions:____________
2.) y = x 4 − x 3 + 2 x 2 − 4 x − 8
Solutions:_____________
3.) y = x 4 + x 2 − 12
Solutions:_______________
Quartic Equations will always have a total of ________ real and imaginary solutions.
6.7
Using the Fundamental Theorem of Algebra
Quintic Equations
Degree:___________
y = x 5 − 2 x 4 + 8 x 2 − 13 x + 6
Quintic Equations will always have a total of ________ real and imaginary solutions.
Write the polynomial function of the least degree that has real coefficients,
the given zeroes and a leading coefficient of 1.
1.) -6,3,5
2.) 2,-2,-6i
3.) 5, 2+3i
6.7
Using the Fundamental Theorem of Algebra
Homework: p. 369 #22-46 even