Polynomial Functions and Complex Zeros

Olivia is solving the equation x^2 - 4x = -8 in her math class. Which statement gives the number and type of solutions for this equation?

• The equation has one real solution.

• The solutions can only be determined by graphing.

• The equation has two real solutions.

• The equation has two nonreal complex solutions.

Imagine Mason is analyzing a polynomial function f(x) = x^4 - 12x + 1 as part of his mathematics project. Including all real and imaginary solutions, how many solutions exist for this function?

• 0

• 1

• 2

• 4

William is analyzing a quadratic polynomial in his math class. Which statement is true about the zeros of this polynomial?

• The polynomial can have a maximum of 2 real zeros.

• There must be at least one complex zero for every real zero.

• If -5 and 4 are zeros, then the conjugates of -5 and 4 must also be zeros.

• If the polynomial has 3 zeros, it must have imaginary coefficients.

Michael is studying quadratic polynomials and is trying to determine which pair of values could not be the roots of a quadratic polynomial. Which pair should he choose?

• 3 and 14

• -5 and -5i

• -2 + 3i and -2 - 3i

• -14 and 14

If a polynomial has zeros at (5 - √2), 7, and 0, what is the lowest possible degree of the polynomial?

• 6

• 5

• 4

• 3

Imagine Luna is working on a complex project involving polynomial equations. She encounters the equation 7x^4 - 5x^3 + 9x^8 + 10x - 6 = 0. How many solutions, including all real and imaginary solutions, does this equation have?

• 6

• 7

• 8

• 9

Scarlett is analyzing the function x^3 + 5x^2 + 21x - 16 in her mathematics project. She needs to determine the number and type of zeros for this function. How many and what type of zeros does the function have?

• 1 imaginary and 2 real solutions

• 2 imaginary and 1 real solutions

• 0 imaginary and 3 real solutions

• 3 imaginary and 0 real solutions

Oliver is studying polynomial functions in his mathematics class. He needs to determine which of the following polynomial functions have exactly 5 roots, including all real and imaginary roots. Can you help him?

• f(x) = 5x^4 + x^3 - x^2 + x - 5

• f(x) = 4x^5 + 3x^4 + 2x^3 + x^2 + x

• f(x) = 5x^6 - 5x^4 + 5x^2 - 5

• f(x) = 4x^4 - 3x^3 + 2x^2 - 1

Rohan is analyzing different graphs for his mathematics project. He needs to identify which graph could represent a polynomial function with a degree of 3. Can you help him?

• Graph A

• Graph B

• Graph C

• Graph D

Anika is analyzing a polynomial graph for her math project. She is particularly interested in the behavior of the graph at its zeros. She asks, 'Which zero will "bounce" off of the x-axis for x²(x - 1)(x + 2)(x + 3)?'

• 0

• 1

• -2

• -3