Quartic function, HELP!?

Question

Which statement is/are true?

a) A quartic function may have 3 or 4 real zeros
b) A quartic function may have 1 or 2 real zeros
c) A quartic function may have no real zero
d) All of the above

Answer 1

D) All of the above. A quartic function may be entirely above or entirely below the x-axis, thus having no real zeroes, like y = x^4 + 1. It may be tangent to the x-axis at one point like y = x^4, having one real zero (which is a quadruple root at x = 0). y = x^4 - 1 has two real zeros at x = -1 and x = 1, and no others. By choosing two distinct factors and then a double factor, I can construct y = (x + 1)(x - 1)x^2 to have exactly three real zeroes: -1, 0, and 1; zero is a double root. And, finally, I can construct a quartic with four real zeroes by taking y = (x -1)(x)(x + 1)(x + 2). It has four real zeroes: -2, -1, 0, and 1. A polynomial of order n can always have any number of real zeroes from 0 to n. In this case, n = 4.

Answer 2

The answer is all of the above.

A quartic function that has 3 real zeros (provided repetitions are counted as non-repeated):
f(x) = (x - 2)^2(x - 3)(x - 4)

A quartic function that has 4 real zeros:

f(x) = x(x - 1)(x + 1)(x - 2)

A quartic function that has one real zero:

f(x) = (x - 1)^4

A quartic function that has two real zeros:

f(x) = x^2 (x - 1)^2

A quartic function that has no real zeros:

f(x) = (x^2 + 4)(x^2 + 1)

I *think* the answer is all of the above.

Answer 3

P(x) = x^4 - 4x^3 + 3x^2 + 4x - 4 First, locate the first spinoff or P'(x) 4x^3 - 12x^2 + 6x + 4 (divide the entire time period by 2) 2x^3 - 6x^2 + 3x + 2 then do the unreal branch procedure 2 - 6 + 3 + 2 l_2_l 4 -4 -2 -------------------- 2 - 2 -a million 0 2x^2 - 2x - a million then use the quadritic formula the solutions must be 2, -0. 37, and1.37 in case you'll use the quadratic formula you would get an answer with an intensive.