$P (x)$ $P(x)$ is a polynomial function. If $P (x^{2}) = (a - b + 2) x^{3} - 2 x^{2} + (2 a + b + 7) x - 20$ $P(x^2) = (a-b+2)x^3 - 2x^2 + (2a+b+7)x - 20$ , what is $P (a + b)$ $P(a+b)$ ?

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$P (x)$ $P(x)$ is a polynomial function. If $P (x^{2}) = (a - b + 2) x^{3} - 2 x^{2} + (2 a + b + 7) x - 20$ $P(x^2) = (a-b+2)x^3 - 2x^2 + (2a+b+7)x - 20$ , what is $P (a + b)$ $P(a+b)$ ?

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Answer 1 · 2016-06-20T12:13:24

$P (x)$ $P(x)$ cannot be a polynomial because $P (x^{2})$ $P(x^2)$ would certainly be an even function.

Explanation:

If $P (x) = n \sum i = 0 a_{i} x^{i}$ $P(x) = sum_{i=0}^na_ix^i$ then

$P (x^{2}) = n \sum i = 0 a_{i} {(x^{2})}^{i} = n \sum i = 0 a_{i} x^{2 i}$ $P(x^2) = sum_{i=0}^na_i(x^2)^i = sum_{i=0}^na_i x^{2i}$

The proposition is not feasible once defined $P (x)$ $P(x)$ as a polynomial.

Answer 2 · 2016-06-20T22:02:22

$P (a + b) = - 12$ $P(a+b) = -12$

Explanation:

$P (x^{2}) = (a - b + 2) x^{3} - 2 x^{2} + (2 a + b + 7) x - 20$ $P(x^2) = (a-b+2)x^3-2x^2+(2a+b+7)x-20$

Since $P (x)$ $P(x)$ is a polynomial function, any powers of $x$ $x$ in $P (x^{2})$ $P(x^2)$ must be even, not odd.

So we require $(a - b + 2) = 0$ $(a-b+2) = 0$ and $(2 a + b + 7) = 0$ $(2a+b+7) = 0$

Adding these two equations together, we get: $3 a + 9 = 0$ $3a+9 = 0$ , hence $a = - 3$ $a=-3$ and $b = - 1$ $b = -1$

$P (x^{2}) = - 2 x^{2} - 20$ $P(x^2) = -2x^2-20$

So:

$P (x) = - 2 x - 20$ $P(x) = -2x-20$

Hence:

$P (a + b)$ $P(a+b)$

$= P ((- 3) + (- 1))$ $= P((-3)+(-1))$

$= P (- 4)$ $= P(-4)$

$= - 2 (- 4) - 20$ $= -2(-4)-20$

$= 8 - 20$ $= 8-20$

$= - 12$ $= -12$

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$P (x)$ $P(x)$ is a polynomial function. If $P (x^{2}) = (a - b + 2) x^{3} - 2 x^{2} + (2 a + b + 7) x - 20$ $P(x^2) = (a-b+2)x^3 - 2x^2 + (2a+b+7)x - 20$ , what is $P (a + b)$ $P(a+b)$ ?

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Explanation:

Explanation:

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P(x)P(x) is a polynomial function. If P(x2)=(a−b+2)x3−2x2+(2a+b+7)x−20P(x^2) = (a-b+2)x^3 - 2x^2 + (2a+b+7)x - 20 , what is P(a+b)P(a+b) ?

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$P (x)$ $P(x)$ is a polynomial function. If $P (x^{2}) = (a - b + 2) x^{3} - 2 x^{2} + (2 a + b + 7) x - 20$ $P(x^2) = (a-b+2)x^3 - 2x^2 + (2a+b+7)x - 20$ , what is $P (a + b)$ $P(a+b)$ ?