Vocab
explain the difference between roots and factors.
Factors are expressions (x-2)(x+6)
roots are x intercepts x=2 x=-6
What is the degree of the polynomial:
A: y=x(x-2)3(x+10)2
B: y=x4-x2+10+x10
A: Degree 6
B: Degree 10 **Careful not in standard form**
Degree 3 polynomial with a positive lead coefficient
roots @ 0 , 3 , -3
ex:
y = x(x+3)(x-3)
If you are shown the graph of a polynomial, how can you tell if the roots are complex or real?
Real roots intersect at the x-axis
Complex roots will not intersect at the x-axis.
What is the complex conjugate of -3+2i
-3-2i
The highest exponent of a polynomial
What is the degree
Which of the following equations is NOT a polynomial? Explain your reasoning or show the work that helped you decide on your answers.
A:y=x2-x+2/x
B: y=x
C: y=(3x-5)(x+2)
D: y=3x4-2x
A:y=x2-x+2/x ** Can't divide by x**
Sketch the graph of the function y=(x+1)(x-3)2(x+4)3
Mark and label the x-intercepts and y-intercepts. What is the degree of this polynomial?
x= -1,3 (bounce), -4 (flat)
y inercept (0,576)
Degree 6 (up)
Can a degree 4 polynomial have no real roots?
YES
Draw Graph
3 turning points
Factor (x2+1)
(x-i)(x+i)
I can be used to find the zeros of a quadratic.
Quadratic Formula
What is the y-intercept of the following polynomial:
y=x4-8x3+2x+8
The y-intercept is when x=0
y intercept is (0,8)
Sketch the graph of y=-2x(x+3)(x-4)2
Mark and label the x-intercepts and y y-intercepts. What is the degree of this polynomial?
Degree 4 (down)
X intercepts: x= - 0, -3, 4 (double)
y int (0,0)
Suppose a cubic function f(x) has 4 + 3i as one of its roots. Can it have a repeated real root? Explain completely.
No.
The complex conjugate would also need to be a root 4-3i.
Thus, the real root can only happen once since a degree 3 has 3 roots.
Solve for the roots of the following function:
y= (x-2)(x2+x+10)
x= -2
x= (-1/2) + (i sqr(39))/2
x=x= (-1/2) - (i sqr(39))/2
When you add or multiply me together, you get a real number.
6+5i and 6-5i
what are complex conjugates
What is the y-intercept of the following polynomial:
y=(x+4)(x+2)2(x+3)
(0,48)
Choose all of the following statements that apply to the graph of
y= -3x(x-5)3(x+2)2
Explain your reasoning or show the work that helped you decide on your answers.
A. The degree is 6.
B. It has negative orientation.
C. It has 6 distinct x-intercepts.
D. The range is all real numbers.
E. It passes through the origin.
F. (0, 2) is one of the roots.
G. Its inverse is a function.
H. The y-intercept is (0, –500).
A,B,D,E
Jeremiah was doing his class work with his team when he and his team members had to write the equation of a parabola that had a complex root of
x = 2 + 5i. Elise said she knew the parabola did not touch the x-axis and the other root was “easy” to find.
a. Why is the parabola not touching the x-axis
b. What is the other root
c. what is the equation of the parabola in standard form. (use your notes from 8-59a to help you!)?
a: complex root
b: 2-5i (complex conjugate)
c: y=(x-2)2+25. --> y=x2-4x+29
Make a sketch of a polynomial function of degree six with a negative leading coefficient, which has four real roots and two complex roots.
Your graph should have end behavior pointing down.
Degree 6 -> 5 turning points
intersect the x-axis 4 times (3 turning points)
should have 2 turning points not intersecting the x-axis.
describes the values of a function at the positive and negative extremes in its domain.
-- when my x values are very very large or very very small (negative).
What is end behavior ?
What is the y-intercept of the following polynomial:
y=-x(x+2)(x-9)2
A polynomial function has the equation y = ax(x − 4)2(x + 2) and goes through the point (5, −35).
What is the value of a? Write the exact equation
y= -1x(x-4)2(x+2)
Re write F(x)= (x2+1)(x2-4) as the product of four linear pairs with complex coefficients
y= (x-i)(x+i)(x-2)(x+2)
B. 2 complex and 4 real
C. 6 complex