Question

for each function, determine whether it is a polynomial function.

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$$\begin{tabular}{|l|c|c|} \\hline \\multirow{2}{*}{function} & \\multicolumn{2}{c|}{is the function a polynomial?} \\\\ \\cline{2-3} & yes & no \\\\ \\hline (a) \\( f(x) = \\frac{1}{x^4} - 8x^9 + 4 \\) & & \\\\ \\hline (b) \\( v(x) = 6\\sqrt{x} \\) & & \\\\ \\hline (c) \\( g(x) = 5x^2(3x + 2)(x - 5) \\) & & \\\\ \\hline (d) \\( u(x) = -x^3 + 6x^6 \\) & & \\\\ \\hline \\end{tabular}$$

Explanation:

Analyze function (a)

$$ f(x) = \frac{1}{x^4} - 8x^9 + 4 = x^{-4} - 8x^9 + 4 $$

The exponent \(-4\) is not a non-negative integer.

Analyze function (b)

$$ v(x) = 6\sqrt{x} = 6x^{1/2} $$

The exponent \(\frac{1}{2}\) is not a non-negative integer.

Analyze function (c)

$$ g(x) = 5x^2(3x+2)(x-5) = 15x^4 - 71x^3 - 50x^2 $$

All exponents are non-negative integers.

Analyze function (d)

$$ u(x) = -x^3 + 6x^6 $$

All exponents are non-negative integers.

Answer:

No.	Function	Is the function a polynomial?
(b)	\(v(x) = 6\sqrt{x}\)	No
(c)	\(g(x) = 5x^2(3x+2)(x-5)\)	Yes
(d)	\(u(x) = -x^3 + 6x^6\)	Yes