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(A) Given $f(x) = \frac{a x^{10} + b x^8 + c x^6 + d x^4 + e x^2 + 12 x + 15}{x}$.Dividing each term by $x$, we get:$f(x) = a x^9 + b x^7 + c x^5 + d

Vedclass · Accepted Answer

$28$

Vedclass · Answer

(A) Given $f(x) = \frac{a x^{10} + b x^8 + c x^6 + d x^4 + e x^2 + 12 x + 15}{x}$.Dividing each term by $x$, we get:$f(x) = a x^9 + b x^7 + c x^5 + d x^3 + e x + 12 + \frac{15}{x}$.Now, consider $f(-x)$:$f(-x) = a(-x)^9 + b(-x)^7 + c(-x)^5 + d(-x)^3 + e(-x) + 12 + \frac{15}{-x}$$f(-x) = -(a x^9 + b x^7 + c x^5 + d x^3 + e x) + 12 - \frac{15}{x}$.Adding $f(x)$ and $f(-x)$:$f(x) + f(-x) = (a x^9 + b x^7 + c x^5 + d x^3 + e x + 12 + \frac{15}{x}) + (-(a x^9 + b x^7 + c x^5 + d x^3 + e x) + 12 - \frac{15}{x})$$f(x) + f(-x) = 12 + 12 = 24$.For $x = 4$:$f(4) + f(-4) = 24$.Given $f(4) = -4$, we have:$-4 + f(-4) = 24$$f(-4) = 24 + 4 = 28$.

If $f(x)$ is a real-valued function defined by $f(x) = \frac{a x^{10} + b x^8 + c x^6 + d x^4 + e x^2 + 12 x + 15}{x}$ for $x \neq 0$ and $f(4) = -4$, then find the value of $f(-4)$.

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