THE FUNDAMENTAL THEOREM OF ALGEBRA

Solve for the roots of x in each of the equations below. x4 - 81 = 0 x4 + 10x2 + 25 = 0 x4 - x2 - 6 = 0

A) x^4-81=0 - x= 3, -3 B) x^4+10x^2+25=0 - x= i square root 5, i square root -5 C) x^4- x^2-6=0 - x= square root 3, square root -3, i square root 2, -i square root 2

Solve the following equation by identifying all of its roots including any imaginary numbers and multiple roots. (x2 - 1)(x2 + 2)(x + 3)(x - 4)(x + 1) = 0

x = 4 x = -3 x= 0.0000 - 1.4142 i x= 0.0000 + 1.4142 i x = 1 x = -1

Which polynomial equations have -i as one of their roots?

x3 + 3x2 + x + 3 = 0 x3 - 6x2 - 16x + 96 = 0

Which of these polynomial equations is of least degree and has -1, 2, and 4 as three of its roots?

x3 - 5x2 + 2x + 8 = 0

Solve for the roots in the equation below. x4 + 3x2 - 4 = 0

x= -1, 1, 2i. -2i

Solve the following equation by identifying all of its roots including all real and complex numbers. In your final answer, include the necessary steps and calculations. Hint: Use your knowledge of factoring polynomials. (x2 + 1)(x3 + 2x)(x2 - 64) = 0

x^5+2x^3)(x^3+2x)(x^5-64x^3)(2x^3-128x) x= i, -i, 0, i square root 2, i square root -2, 8, -8

Write a polynomial equation of degree 3 such that two of its roots are 2 and an imaginary number.

y = (x - 2)(x - i)(x + i)

Which polynomial equation of least degree has -2, -2, 3, and 3 as four of its roots?

(x + 2)2(x - 3)2 = 0

Solve for the roots in the equation below. In your final answer, include each of the necessary steps and calculations. Hint: Use your knowledge of polynomial division and the quadratic formula. x3 - 27 = 0

(x+3)(x^2-3x+9)=0

Find all the roots of the equation x4 - 2x3 + 14x2 - 18x + 45 = 0 given that 1 + 2i is one of its roots.

1+3=5x3, 5xp, =3 roots because of all of its roots divid

A. How many real roots does the function have? B. Complete the equation of the graphed function. y= (x - )(x + )2

2, 2, 4

Which is a possible number of distinct real roots for a cubic function? Select all that apply.

2, 3, 0

Based on the Fundamental Theorem of Algebra, how many complex roots does each of the following equations have? Write your answer as a number in the space provided. For example, if there are twelve complex roots, type 12. x(x2 - 4)(x2 + 16) = 0 has a0 complex roots (x 2 + 4)(x + 5)2 = 0 has a1 complex roots x6 - 4x5 - 24x2 + 10x - 3 = 0 has a2 complex roots x7 + 128 = 0 has a3 complex roots (x3 + 9)(x2 - 4) = 0 has a4 complex roots

2, 4, 6, 7, 5

How many roots does the equation -2x3 = 0 have?

3

A polynomial equation has only imaginary roots and no real root. Which options CANNOT be the degree of that polynomial? (Select all that apply.)

3, 7

What is the least possible degree of a polynomial that has the root -3 + 2i, and a repeated root -2 that occurs twice?

4

What is the least possible degree of a polynomial that has roots -5, 1 + 4i, and -4i?

5