I thought of these for about 20 minutes and still don't know the answers.
#1: Why must every polynomial equation (with real coefficients) of degree 3 have at least 1 real root?
#2: Explain why x^4 + 6x^2 + 2 = 0 has no rational roots
Thank you so much to whoever is able to help me out with this!
#1: Why must every polynomial equation (with real coefficients) of degree 3 have at least 1 real root?
#2: Explain why x^4 + 6x^2 + 2 = 0 has no rational roots
Thank you so much to whoever is able to help me out with this!
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#1: Because a polynomial of degree 3 has two points of concavity, one pointing up and one pointing down. What this really means is that one end of the curve goes to negative infinity and the other end goes to positive infinity. And because zero exists between negative and positive infinity, the curve must pass through zero, meaning is has to have at least one real root.
If you are curious, this applies to all polynomials with odd degrees (1, 3, 5, etc). So a polynomial of degree 1 (a straight line) either passes through zero (in which it has one root), or it lies on the axis in which all points are roots.
#2: If you take the derivative of the equation "x^4 + 6x^2 + 2 = y" and set it to zero, you can solve for the concavity points.
d/dx ( x^4 + 6x^2 + 2 ) = 4x^3 + 12x = 0
Factoring out the x we get:
x (4x^2 + 12) = 0
so one of the concavities is at zero, and the other is at:
4x^2 + 12 = 0
4x^2 = -12
x^2 = -3
x = sqrt(-3) = imaginary number
which means that there is really only one concavity (at x = 0).
Now, we need to determine if the concavity is up or down at x = 0. To do this, you need to solve for the second derivative at x = 0.
The second derivative is the derivative of the first derivative, or:
d/dx ( 4x^3 + 12x ) = 12x + 12
If we solve for x = 0, then we get the value 12, which is a positive concavity. Therefore we know the function curves upwards. This also means that the point at x = 0 is a minimum.
If we plug x=0 into the very first equation (x^4 + 6x^2 + 2 = y), then we find that y = 2.
So since we know that the lowest value of the curve is 2, then the curve must never pass through zero and therefore has no roots.
Hopefully that answers your question. I apologize if you are not allowed to use calculus... For the life of me I can't remember how to answer the question with just algebra. So if this answer doesn't work for you, just send me a message and I can try again.
If you are curious, this applies to all polynomials with odd degrees (1, 3, 5, etc). So a polynomial of degree 1 (a straight line) either passes through zero (in which it has one root), or it lies on the axis in which all points are roots.
#2: If you take the derivative of the equation "x^4 + 6x^2 + 2 = y" and set it to zero, you can solve for the concavity points.
d/dx ( x^4 + 6x^2 + 2 ) = 4x^3 + 12x = 0
Factoring out the x we get:
x (4x^2 + 12) = 0
so one of the concavities is at zero, and the other is at:
4x^2 + 12 = 0
4x^2 = -12
x^2 = -3
x = sqrt(-3) = imaginary number
which means that there is really only one concavity (at x = 0).
Now, we need to determine if the concavity is up or down at x = 0. To do this, you need to solve for the second derivative at x = 0.
The second derivative is the derivative of the first derivative, or:
d/dx ( 4x^3 + 12x ) = 12x + 12
If we solve for x = 0, then we get the value 12, which is a positive concavity. Therefore we know the function curves upwards. This also means that the point at x = 0 is a minimum.
If we plug x=0 into the very first equation (x^4 + 6x^2 + 2 = y), then we find that y = 2.
So since we know that the lowest value of the curve is 2, then the curve must never pass through zero and therefore has no roots.
Hopefully that answers your question. I apologize if you are not allowed to use calculus... For the life of me I can't remember how to answer the question with just algebra. So if this answer doesn't work for you, just send me a message and I can try again.
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1) because it must cross the X axis somewhere (to get from negative to positive Y values)
2) because it's "floating above" the X axis without crossing it
2) because it's "floating above" the X axis without crossing it