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10
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1
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A determinant involving a polynomial is $0$
Let $n \geq 2$ and $f:\mathbb{R} \to \mathbb{R}, \: f(x)=(x-x_1)(x-x_2)\dots(x-x_n)$ where $x_1,\dots, x_n$ are distinct real numbers. The matrix $A=(a_{ij})_{1 \leq i,j \leq n}$ is defined as follows:...
- 1,837
2
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3
answers
116
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Formula for $1/f(x)$ where $f$ is a polynomial
Let $f$ be a polynomial having $n$ distinct real roots:
$$f(x)=(x-x_1)(x-x_2)\dots(x-x_n)$$
Prove that $$\frac{1}{f(x)}=\sum_{k=1}^n \frac{1}{f'(x_k)(x-x_k)}, \: \forall x \in \mathbb{R} - \{x_1,...
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