Question

For 2x–5x^3 + 12 = 0
, state the number of complex roots, the possible number of real roots, and the possible rational roots.
would there be 5 complex?
@mathmale

Answer 1

@mathstudent55

Answer 2

is the equation \[2x-5x^3+12 = 0\]

Answer 3

Assuming it is, you can determine the number of total roots by finding the largest exponent in any of the terms of the polynomial. You will always have that many solutions to a polynomial, no more, no less. Sometimes a solution will repeat, but they count just the same.

In a polynomial with only real coefficients, such as this one, complex solutions always come in complex conjugate pairs, of the form \(a \pm bi\) where \(i = \sqrt{-1}\). If you had a single complex solution, then you would have one or more complex coefficients.

From that information, what can you tell me about the solutions to this polynomial?

Answer 4

yes thaat is the equation

Answer 5

5 complex and 1 3 or 5 real roots @whpalmer4

Answer 6

what is the largest exponent you see there?

Answer 7

wait sorry 3

Answer 8

can you write the term that has a 5 for an exponent?

Answer 9

there we go :-)

Answer 10

number in front of a variable = coefficient
number raised up above = exponent

Answer 11

so would there be 3 complex

Answer 12

Did you read my two paragraphs? What did I say about complex solutions?

Answer 13

they come in pairs

Answer 14

so 2?

Answer 15

well, either 0 or 2, we don't know yet. but not 3, and not 1.

Answer 16

do you know what the shape of the graph of an equation with an x^3 in it looks like?

Answer 17

yes i just did it on my calc

Answer 18

if we just have \(x\), we get a straight line
if we have an \(x^2\), we get a parabola of some sort
if we have an \(x^3\), we get sort of a loopy n shape

(plus all the versions can be flipped upside down, left for right, etc.)

Answer 19

solutions are the points where the line or curve crosses the x-axis. the cubic that I drew as my example has 3 real solutions because it crosses the x-axis 3 times.