ediculeN
2020-10-28
Answered

Need to calculate:The factorization of the polynomial ${a}^{2}+ac+a+c$

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curwyrm

Answered 2020-10-29
Author has **87** answers

Formula used:

The factors of a polynomial can be found by taking a common factor and this method is called factor by grouping,

Or,

Calculation:

Consider the polynomial

This is a four term polynomial, factorization of this polynomial can be find by factor by grouping as,

As,

The factorization of the polynomial

Check the result as follows:

Thus, the factorization of the polynomial

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I want to approximate $\mathrm{tanh}$ by a low-degree rational function of form

$r(x)=\frac{p(x)}{q(x)}=\frac{{p}_{2}{x}^{2}+{p}_{1}x+{p}_{0}}{{q}_{1}x+{q}_{0}}$

such that the ${L}^{2}$ norm over the fixed interval $[{x}_{0},{x}_{1}]$ is small:

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That is, I would like to solve for the coefficients ${p}_{2},{p}_{1},{p}_{0},{q}_{1},{q}_{0}$. Note that $\mathrm{d}\mathrm{e}\mathrm{g}(p)\le 2$ and $\mathrm{d}\mathrm{e}\mathrm{g}(q)\le 1$.

I don't know where to begin, so I'm looking for suggestions on how to approach this problem; pointers to relevant numerical methods would also be greatly appreciated

$r(x)=\frac{p(x)}{q(x)}=\frac{{p}_{2}{x}^{2}+{p}_{1}x+{p}_{0}}{{q}_{1}x+{q}_{0}}$

such that the ${L}^{2}$ norm over the fixed interval $[{x}_{0},{x}_{1}]$ is small:

$I(p,q)=\frac{1}{2}{\int}_{{x}_{0}}^{{x}_{1}}{(r(x)-\mathrm{tanh}(x))}^{2}\mathrm{d}x.$

That is, I would like to solve for the coefficients ${p}_{2},{p}_{1},{p}_{0},{q}_{1},{q}_{0}$. Note that $\mathrm{d}\mathrm{e}\mathrm{g}(p)\le 2$ and $\mathrm{d}\mathrm{e}\mathrm{g}(q)\le 1$.

I don't know where to begin, so I'm looking for suggestions on how to approach this problem; pointers to relevant numerical methods would also be greatly appreciated

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