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Olympiad workbook

Exam: Olympiad workbook
Subject: IOQM
Chapter: Quadratic Equation

Olympiad workbook · IOQM · Quadratic Equation

Suppose that P is the polynomial of least degree with integer...

Suppose that P is the polynomial of least degree with integer coefficients such that

\mathrm{P}(\sqrt{7}+\sqrt{5})=2(\sqrt{7}-\sqrt{5})

. Find

\mathrm{P}(2)

.

Correct answer

40

Step-by-step solution

Let

\sqrt{7}+\sqrt{5}=\mathrm{x}

then

\begin{aligned} & P(x)=(\sqrt{7}+\sqrt{5})(\sqrt{7}-\sqrt{5})(\sqrt{7}-\sqrt{5}) \quad \text { since } 2=(\sqrt{7}+\sqrt{5})(\sqrt{7}-\sqrt{5}) \\ & =(\sqrt{7}+\sqrt{5})(\sqrt{7}-\sqrt{5})^2 \\ & =(\sqrt{7}+\sqrt{5})\left(24-(\sqrt{7}+\sqrt{5})^2\right) \\ & P(x)=x\left(24-x^2\right) \\ & P(2)=2(24-4) \\ & \hline \end{aligned}

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