In Problems 1-4, determine whether the given function is a polynomial function. $$ \begin{array}{ll} ext { 1. } f(x)=x^{2}-x^{4}+4 & ext { 2. } f(x)=\frac{x^{3}+7 x-3}{3} \ ext { 3. } g(x)=\frac{1}{x^{2}+2 x+1} & ext { 4. } g(x)=2^{-3} x^{3}\end{array} $$

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In Problems 1-4, determine whether the given function is a polynomial function. $$ \begin{array}{ll}	ext { 1. } f(x)=x^{2}-x^{4}+4 & 	ext { 2. } f(x)=\frac{x^{3}+7 x-3}{3} \ 	ext { 3. } g(x)=\frac{1}{x^{2}+2 x+1} & 	ext { 4. } g(x)=2^{-3} x^{3}\end{array} $$

UpStudy AI · Accepted Answer

**1. $$ f(x)=x^2-x^4+4 $$**  
All terms in the function are monomials with nonnegative integer exponents and constant coefficients. Therefore,  
$$
f(x)=x^2-x^4+4
$$  
is a polynomial.

**2. $$ f(x)=\frac{x^3+7x-3}{3} $$**  
Since division by a nonzero constant preserves the polynomial structure,  
$$
f(x)=\frac{1}{3}(x^3+7x-3)
$$  
is a polynomial.

**3. $$ g(x)=\frac{1}{x^2+2x+1} $$**  
The denominator can be factored as:  
$$
x^2+2x+1=(x+1)^2.
$$  
Thus,  
$$
g(x)=\frac{1}{(x+1)^2}.
$$  
This is not a polynomial because it involves taking the reciprocal of a polynomial expression; the result is a rational function in which the denominator is not $$1$$.

**4. $$ g(x)=2^{-3}x^3 $$**  
Notice that $$2^{-3}=\frac{1}{8}$$, so  
$$
g(x)=\frac{1}{8}x^3.
$$  
This is a monomial with a nonnegative integer exponent and a constant factor. Hence, it is a polynomial.
1. $$ f(x)=x^2-x^4+4 $$ is a polynomial. 2. $$ f(x)=\frac{x^3+7x-3}{3} $$ is a polynomial. 3. $$ g(x)=\frac{1}{x^2+2x+1} $$ is not a polynomial. 4. $$ g(x)=2^{-3}x^3 $$ is a polynomial.

In Problems 1-4, determine whether the given function is a
polynomial function.

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