Question

Which of the following is/are polynomial functions?

• A. $g\left(x\right)=(1+2x)(x+\frac{4}{x})$
• B. $p\left(x\right)=5^{x^2+x+1}$
• C. $f\left(x\right)=(x+1)(x+2)(x-1)(x-2)$
• D. $h\left(x\right)=x^3+(\frac{e}{e+1})x^2-\left(\ln 6\right)x$

Answer 1

Answer: (C), (D)

$h\left(x\right)=x^3+(\frac{e}{e+1})x^2-\left(\ln 6\right)x$

Polynomial functions are defined by
$F\left(x\right)=a_0{x}^n+a_1{x}^{n-1}+a_2{x}^{n-2}+\cdots +a_{n-1}x+a_n$
where $a_i\in R,n\in W,a_0\ne 0,\forall x\in R$
And the degree of $f\left(x\right)$ will be $n$.

We have
$f\left(x\right)=(x+1)(x+2)(x-1)(x-2)$
$f\left(x\right)=x^4-3x^2+2$
Which is a polynimal function.

Taking,
$g\left(x\right)=(1+2x)(x+\frac{4}{x})$
$g\left(x\right)=(1+2x)(x+4x^{-1})$
Since $-1\notin W$, So it is not a polynomial.

Consider,
$h\left(x\right)=x^3+(\frac{e}{e+1})x^2-\left({\log }_{10}6\right)x+27$
we can clearly observe that all the coefficients are real numbers and powers of variable $x$ are whole number.
Hence it is a polynomial.

Also, $p\left(x\right)=5^{x^2+x+1}$
Obviously not a polynimial function, since it's an exponential function.