# Questions tagged [discriminant]

Discriminant of a polynomial $\;P\left(x\right) = a_{0} + a_{1}x + a_{2}x^{2} + \dots + a_{n}x^{n} \neq 0\,$ is defined as \begin{align} \Delta &= a_{n}^{2n-2}\prod_{ i < j } \big( r_i - r_j \big)^{2} = \left(-1\right)^{n\left(n-1\right)/2} a_{n}^{2n-2}\prod_{ i \neq j } \big( r_i - r_j \big) \end{align} where $\,r_1,\dots,r_n\,$ are roots of $P\left(x\right)$ (counting multiplicity)

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### how would i find the domain for this prove question using the discriminant

How would I prove that 3x^2-x+7=0 goes for all real x I've already tried discriminant method b^2-4ac but it's still incorrect and using the quadratic formula gives me an error.
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### Discriminant of depressed cubic [duplicate]

If the cubic equation $x^3+px+q$ has roots $\alpha , \beta , \gamma$ then we know that $\alpha + \beta + \gamma =0$, $\alpha \beta + \alpha \gamma + \beta \gamma =p$ and $\alpha \beta \gamma =-q$. ...
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### What is the number of solution of equation of locus when a straight line is tangent to the locus.

Find the values of t if a straight line y=t is the tangent to the locus of point P , $$x^2+y^2+4x-6y-3=0$$ where t is a constant. This is an exam question from a school. The solution in the answer ...
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### Discriminants of quartic fields are $k$-power free for some $k$?

Let $K/\mathbb{Q}$ be a quadratic number field. Then the discriminant of $K$ is either $d$ or $4d$ for some squarefree integer $d$, so the discriminant is never divisible by an odd square. I am ...
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### Hungerfords definition of the discriminant

Hungerfords Algebra Definition 4.4: Let $K$ be a field with char$K \neq 2$ and $f \in K[x]$ be a polynomial of degree $n$ with $n$ distinct roots $u_1,..u_n$ in some splitting field $F$ of $f$ over $K$...
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### What is the discriminant of $\Phi_{2}(X)$ over $\Bbb Q$?

What is the discriminant of $\Phi_{2^n}(X)$ over $\Bbb Q$ for $n=1$ since $\Phi_2(X)=X+1$ has only one root $-1$? I have calculated $disc(\Phi_{2^n}(X))$ for $n\geq 2$ which matches exactly with what ...
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### Discriminant Problems

We all know that the discriminant is the part $b^2-4ac$ of the equation $$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$ that we use to find the roots of a quadratic equation eg: $ax^2+bx+c=0$ or the part in a ...
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### $\displaystyle |g(x)| \leq |f(x)|$ for polynomials

I have got a polynomial: $$f(x) = x^4 - 5x^2 + 5$$ And the condition for a polynomial $g(x)$: $$\forall x \in \mathbb{R}, |g(x)|\leq |f(x)|$$ Prove that $f(x) = a \cdot g(x)$ It's quite easy to see ...
### $\mathbb{Q}(\alpha)$ extension of degree 3 is galois over $\mathbb{Q}$ if and only if discriminant of minimal polynomial of $\alpha$ is square.
I supposed that $\alpha$, $\beta$ and $\gamma$ are the roots of the minimal polynomial in its splitting field. So the discriminant is $(\alpha-\beta)^2(\alpha-\gamma)^2(\beta-\gamma)^2$. If every ...