query: Show a real polynomial has at least one zero
positive_passages.docid: 155245
positive_passages.text: root of an odd degree polynomial
negative_passages.docid: 154507
negative_passages.text: If all the roots of a polynomial P(z) have negative real parts, prove that all the roots of P'(z) also have negative real parts
negative_passages.docid: 73478
negative_passages.text: Show that the equation $a_1e^{\alpha_1x} + a_2e^{\alpha_2x} + \cdots + a_ne^{\alpha_nx} = 0$ has at most $n - 1$ real roots.
negative_passages.docid: 61500
negative_passages.text: If f is continous and exist real numbers show that $f(x)=0$
negative_passages.docid: 154393
negative_passages.text: Roots of polynomial with real coefficients appear in conjugate pairs.
negative_passages.docid: 94682
negative_passages.text: Polynomial: prove there is no double root
negative_passages.docid: 103340
negative_passages.text: Show that this entire function is polynomial.
negative_passages.docid: 6722
negative_passages.text: to show $f$ is a polynomial in $z$
negative_passages.docid: 30860
negative_passages.text: Prove that $f(x)=0$ in $[-1,1]$.
negative_passages.docid: 148910
negative_passages.text: Multivariate polynomials have infinitely many zeroes
negative_passages.docid: 55119
negative_passages.text: Proof that the coefficients of a polynomial are real
negative_passages.docid: 112334
negative_passages.text: Real roots of a polynomial of real co-efficients , with the co-efficients of $x^2 , x$ and the constant term all $1$
negative_passages.docid: 14836
negative_passages.text: A polynomial sequence
negative_passages.docid: 100862
negative_passages.text: Polynomials with rational zeros
negative_passages.docid: 109102
negative_passages.text: Determine all real polynomials with $P(0)=0$ and $P(x^2+1)=(P(x))^2+1$
negative_passages.docid: 139614
negative_passages.text: Polynomial bounded real part of an entire function
negative_passages.docid: 76482
negative_passages.text: Closure of the set of degree $n$ polynomials with $n$ real roots