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# review problem blegh

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Find a monic cubic polynomial $$P(x)$$  with integer coefficients such that $$P(\sqrt[3]{2} + 1) = 0.$$
(A polynomial is monic if its leading coefficient is 1.)

Guys, I'm really stuck and its a review problem (can't skip it). Can anyone help? Thanks yall

Apr 24, 2020

#1
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Find a monic cubic polynomial $$P(x)$$ with integer coefficients such that $$P(\sqrt[3]{2} + 1) = 0$$.
(A polynomial is monic if its leading coefficient is $$1$$.)

$$\text{Let a=\sqrt[3]{2}\quad so \quad\mathbf{a^3=2} } \\ \text{Let b=1 } \\ \text{Let x = a+b \quad or \quad \mathbf{a = x-b} }$$

$$\begin{array}{|rcll|} \hline \mathbf{a} &=& \mathbf{x-b} \\\\ a^3 &=& (x-b)^3 \\ a^3 &=& x^3-3x^2b+3xb^2-b^3 \quad | \quad \quad\mathbf{a^3=2},\ \mathbf{b=1} \\ 2 &=& x^3-3x^2+3x-1 \\ \mathbf{x^3-3x^2+3x-3} &=& \mathbf{0} \\ \mathbf{P(x)} &=& \mathbf{x^3-3x^2+3x-3} \\ \hline \end{array}$$

Apr 24, 2020
#2
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thank you SOO much!!

Guest Apr 24, 2020