Cubic With 3 Positive Roots

I think, that by Descartes' rule of signs, the signs of the coefficients must interleave for the solutions to be positive, so for the quadratic equation $a$ and $c$ must have equal signs and $b$ must have opposite sign; similarity for the cubic equation, $a$ and $c$ must have equal signs, and $b$ and $d$ must have equal signs but opposite from $a$ and $c$.

Now, the second condition is that the discriminant must be greater than 0, that is because the discriminant is just the product of the squaresof every possible difference of two roots at time, all that multiplied by $a^{2n-2}$, where $a$ is the leading coefficient. For example, for a second degree equation: $D=a^2 (x_1-x_2)^2$, and for a third degree equation: $D=a^4 (x_1-x_2)^2 (x_1-x_3)^2 (x_2-x_3)^2$. It's clearly that if at least one root is repeated, the whole discriminant will be 0. And if there are a pair of complex roots it will be negative. But, these discriminants now in terms of the coefficients are: $D=b^2-4ac$ and $D=b^2 c^2 - 4ac^3 - 4b^3 d + 18abcd - 27a^2 d^2$.So, in both cases, $D>0$.

I think that in these grades these are the sufficient and necesary conditions for all roots to be distinct and positive, but in higher degrees I'm not sure. The proofs of how to obtain the discrimimants are a little hard and I will not give them, also I'm writing this from my phone, and it's a pain, so if there are any mistakes, I'm sorry.

For the quadratic equation, 2 conditions arise-

If $a>0$, then $c>0$ and $b^{2} - 4ac>0$. Also, $\dfrac{-b}{2a}>0$ which means that $b<0$

If $a<0$, then $c<0$ and $b^{2} - 4ac>0$. Also, $\dfrac{-b}{2a}>0$ which means that $b>0$

For the cubic equation, 2 conditions arise-

Let $f(x) = ax^{3} + bx^{2} + cx + d$

If the cubic has 3 roots, then $f'(x)$ has $\textbf{2 positive real roots}$. Let they be $p,q$. The conditions for it can be found from above.

For three roots,

$f(p) . f(q) <0$

Assuming that the condition is satisfied, then 2 cases arise-

If $a<0$, then $d>0$

If $a>0$, then $d<0$

This is with regard to second part, although the approach could also be used for quadratic case. The general cubic equation is

$a{ x }^{ 3 }+b{ x }^{ 2 }+cx+d=0$

Convert the general cubic equation into depressed cubic.Thus

$t^{3}-mt-c=0$

with $t=x+b/(3a)$

Then we may consider this to be the intersection of two curves namely $y=t^{3}$and a straight line $y=mt+c$. Now plotting these two curves one clearly sees that for 3 real roots the slope m of the line must be positive.Thus condition one is

$m>0$

This is equivalent to the condition that the derivative of the cubic polynomial must have two roots.Next if there were two repeated roots then the slope of the line and tangent are equal. Equating both we get

${ t }_{ 0 }=\pm \sqrt { \frac { m }{ 3 } }$ .

Substituting t0 back into the straight line equation we get a value for the y-intercept for this special case and noting that the intercept $\left| c \right|$ must be less than this for 3 unequal real roots to exist.Thus we have the condition two

$\left| c \right| \le \quad \frac { 2m }{ 3 } \sqrt { \frac { m }{ 3 } }$

This is equivalent to the discriminant condition of the cubic equation. Next we turn to the question of positive roots. This is equivalent to

$t>\frac { b }{ 3a }$

Now we note that if the depressed cubic has 3 real roots then by necessity it will have one negative root (since sum of roots is zero in depressed cubic & by same logic it will also atleast have one positive root). So for positive real roots of the original cubic we need to have

$\frac { b }{ a } <0$

This third condition is only necessary not sufficient.Next we ensure that the smallest root of depressed cubic is greater than b/3a . For this we suppose a straight line of slope m passes through the point $\left( \frac { b }{ 3a } ,{ \left( \frac { b }{ 3a } \right) }^{ 3 } \right)$ then the y intercept of this line comes out to be

${ c }_{ 0 }={ \left( \frac { b }{ 3a } \right) }^{ 3 }-m\frac { b }{ 3a }$

This c0 must be lesser than y intercept of the straight line for roots of depressed cubic to be greater than b/3a.Thus condition four is

${ c }_{ 0 }<c$.

Once all the four conditions are satisfied simultaneously we will have positive unequal roots for the original cubic.

if x and y are real numbers satisfying x^3-3x^2+5x-17=0 and y^3-3y^2+5y+11=0 then the value of x+y=______

1. The discriminant ( b^2 - 4ac) should be positive and signs must be alternate.

This makes me think about the problems I posted about Brilli the Ant playing the games against Brian Till.

guys, I hate to rain on your parade but can you elaborate this much clearer and simpler for me because I can't understand a word you're typing. please???

1. Discriminant ( b^2-4ac) >0

b sqred - 4ac>0

Cubic Equation x³+ax²+bx+c=0 with real coefficients a,b,c have ONLY REAL ROOTS iff

1) (2a³-9ab+27c)² ≤ 4(a²-3b)³

2) a² ≥ 3b

"+" to be 3 POSITIVE must be by Vietta

3) a<0

4) b>0

5) c<0

a and c having same sign b is different

What is the meaning of all these things i.e possible real roots and all ,these thing are not taught in my school