Let Px be a polynomial satisfying limx-x3Pxx6-3x2-7-2- If P1 - 2- P3 - 10 and P5 - 26- then the value of P2-P0-10 is

Let f(x)=P(x) (x^2+1) Then, x=1,3,5 are roots of f(x). Therefore, we can write f(x) as A(x 1)(x 3)(x 5) nbsp; ⇒ P(x)=A(x 1)(x 3)(x 5)+(x^2+1) Also, P(x) shoul ...

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Byju's Answer

Standard XII

Mathematics

Finding Derivative of a Function

Let Px be a p...

Question

Let $P (x)$ be a polynomial satisfying $lim x \to \infty \frac{x^{3} P (x)}{x^{6} + 3 x^{2} + 7} = 2.$ If $P (1) = 2,$ $P (3) = 10$ and $P (5) = 26,$ then the value of $\frac{P (2) + | P (0) |}{10}$ is

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Solution

Let $f (x) = P (x) - (x^{2} + 1)$
Then, $x = 1, 3, 5$ are roots of $f (x) .$
Therefore, we can write $f (x)$ as $A (x - 1) (x - 3) (x - 5)$
$\Rightarrow P (x) = A (x - 1) (x - 3) (x - 5) + (x^{2} + 1)$
Also, $P (x)$ should be of degree $3.$

We have,
$lim x \to \infty \frac{x^{3} P (x)}{x^{6} + 3 x^{2} + 7} = 2$
$\Rightarrow A = 2$
$\Rightarrow P (x) = 2 (x - 1) (x - 3) (x - 5) + x^{2} + 1$
Now, $P (2) = 11, P (0) = - 29$
$∴ \frac{| P (0) | + P (2)}{10} = 4$

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Let P(x) be a polynomial satisfying limx→∞x3P(x)x6+3x2+7=2. If P(1)=2, P(3)=10 and P(5)=26, then the value of P(2)+|P(0)|10 is

Let $P (x)$ be a polynomial satisfying $lim x \to \infty \frac{x^{3} P (x)}{x^{6} + 3 x^{2} + 7} = 2.$ If $P (1) = 2,$ $P (3) = 10$ and $P (5) = 26,$ then the value of $\frac{P (2) + | P (0) |}{10}$ is