Intereting Posts

Proving $a^ab^b + a^bb^a \le 1$, given $a + b = 1$
References mentioning the relationship between cumulants of uniform distribution and the Bernoulli numbers?
Characterizing continuous functions based on the graph of the function
How prove $\frac{a}{11a+9b+c}+\frac{b}{11b+9c+a}+\frac{c}{11c+9a+b}\le\frac{1}{7}$
Tensor Calculus
Finite Group with $n$-automorphism map
Proving that additive groups are isomorphic $(n\mathbb{Z}$ and $m\mathbb{Z}$) or not ($\mathbb{Q}$ and $\mathbb{Z}$)
Meaning of measure zero
Definition Of Symmetric Difference
Are all values of the totient function for composite numbers between two consecutive primes smaller than the totient values of those primes?
Generalization of the argument principle
2 color theorem
Help with Cramer's rule and barycentric coordinates
$(A\cup B)\cap C = A\cup(B\cap C)$ if and only if $A\subset C$
Binomial rings closed under colimits?

Given a cubic polynomial with real coefficients of the form $f(x) = Ax^3 + Bx^2 + Cx + D$ $(A \neq 0)$ I am trying to determine what the necessary conditions of the coefficients are so that $f(x)$ has exactly three distinct real roots. I am wondering if there is a way to change variables to simplify this problem and am looking for some clever ideas on this matter or on other ways to obtain these conditions.

- Isomorphism between quotient rings over finite fields
- Proving that $\sum_{k=1}^{\infty} \frac{3408 k^2+1974 k-720}{128 k^6+480 k^5+680 k^4+450 k^3+137 k^2+15 k} = \pi$
- Product of two primitive polynomials
- Primitive polynomials
- The Uniqueness of a Coset of $R/\langle f\rangle$ where $f$ is a Polynomial of Degree $d$ in $R$
- Factorize polynomial over $GF(3)$
- Is this an equivalent statement to the Fundamental Theorem of Algebra?
- Proving that a polynomial is irreducible over a field
- Define $f(x),g(x)\in \mathbb{R}$. Prove $f(x)=g(x)$.
- roots of minimal and characteristic polynomial

Suppose that (including multiplicity) the roots of $$f(x) = A x^3 + B x^2 + C x + D,$$ $A \neq 0$ are $r_1, r_2, r_3$. Then, inspection shows that the quantity

$$D(f) := A^4 (r_3 – r_2)^2 (r_1 – r_3)^2 (r_2 – r_1)^2,$$

called the *(polynomial) discriminant* of $f$, vanishes iff $f$ has a repeated root. (The coefficient $A^4$ is unnecessary for the expression to enjoy this property, but among other things, its inclusion makes the below formula nicer.) On the other hand, with some work (say, by expanding and using Newton’s Identities and Vieta’s Formulas) we can write $D(f)$ as a homogeneous quartic expression in the coefficients $A, B, C, D$:

$$D(f) = -27 A^2 D^2 + 18 ABCD – 4 A C^3 – 4 B^3 D + B^2 C^2.$$

It turns out that $D$ gives us the finer information we want, too: $f$ has three distinct, real roots iff $D(f) > 0$ and one real root and two conjugate, nonreal roots iff $D(f) < 0$.

It’s apparent that one can generalize the notion of discriminant to polynomials $p$ of any degree $> 1$, producing an expression homogeneous of degree $2(\deg p – 1)$ in the polynomial coefficients. In each case, up to a constant that depends on the degree and the leading coefficient of $f$, $D(f)$ is equal to the resultant $R(f, f’)$ of $f$ and its derivative $f'(x) = 3 A x^2 + 2 B x + C$.

By making a suitable affine change of variables $x \rightsquigarrow y$, by the way, one can transform the given cubic to the form

$$\tilde{f}(y) = y^3 + P y + Q$$ (which does not change the multiplicity of roots), and for a cubic polynomial in this form the discriminant has the simple and well-known form

$$-4 P^3 – 27 Q^2.$$

By solving $f'(x)=0$, you can find out the two turning points of the cubic. Suppose the solutions are $x_1$ and $x_2$. In fact, we have

$$

x_{1,2} = \frac{-B \pm \sqrt{B^2-3AC}}{3A}

$$

If $B^2-3AC <0$, then $x_1$ and $x_2$ are imaginary, so there are no turning points, and the cubic has only one real root.

If $x_1$ and $x_2$ are real, the cubic has three distinct roots iff $f(x_1)$ and $f(x_2)$ are non-zero and have opposite sign.

There are numerous corner cases to worry about, but that’s the basic idea.

**Edit:** Following up on the comment above, the Wikipedia page says that the nature of the roots can be determined by examining the discriminant:

$$

\Delta = 18ABCD – 4B^3D + B^2C^2 -4AC^3 – 27A^2D^2

$$

The cubic has three distinct real roots iff $\Delta > 0$. This is a nice tidy criterion, but my discussion above may still be useful because it provides some geometric insight.

For simplicity assume $A>0$ (you can easily get analogous conditions for $A<0)$. Then the first turning point will be a maximum and the second one will be a minimum. Then the 3 distinct roots:

(i) the least root should be earlier than the point where maximum is attained

(ii) the middle root should be between the maximum and the minimum

(iii) the largest root would be bigger than the minimum.

This translates to maximum should be attained at a negative number, minimum at a positive number.

You can now express the above statement into a condition on the coefficients of $f’x$.

- Matrices – Conditions for $AB+BA=0$
- Find the splitting field of $x^3-1$ over $\mathbb{Q}$.
- Connected metric spaces with at least 2 points are uncountable.
- Application of maximum modulus principle
- What is the average rotation angle needed to change the color of a sphere?
- Cauchy's argument principle, trouble working simple contour integral
- Frobenius Morphism on Elliptic Curves
- square root of symmetric matrix and transposition
- Numbers ending with 0
- Example of a Non-Abelian Group
- How can I show that $\sqrt{1+\sqrt{2+\sqrt{3+\sqrt\ldots}}}$ exists?
- Endpoint-average inequality for a line segment in a normed space
- Why are polynomials defined to be “formal”?
- positive definite quadratic form
- Find all positive integers $n$ such that $\phi(n)=6$.