Property 1
As each polynomial is a function, all the polynomials provide the same properties of functions.
In real numbers the value of a polynomial can be found by putting the given value in the place, the same way it is done in factors.
1.
\(P(x)=3x+1\)
\(\Rightarrow P(2)=?\)
2.
\(P(x)=x^2-3x\)
\(\Rightarrow P(3)=?\)
3.
\(P(x)=2x^2-3x\)
\(\Rightarrow 2P(1)=?\)
4.
\(P(x)=3x+k\)
\(P(1)=7\)
\(\Rightarrow k=?\)
5.
\(P(x)=x^2-2x+k\)
\(P(1)=10\)
\(\Rightarrow k=?\)
6.
\(P(x+4)=3x^2-2x\)
\(\Rightarrow P(6)=?\)
7.
\(P(2x+1)=3x^2-2\)
\(\Rightarrow P(5)=?\)
8.
\(P(x^2)=2x^4-2x^2\)
\(\Rightarrow P(3)=?\)
Property 2
To define a function as a polynomial, exponents of x should be natural number.
\[ \mathbb{N}=\{0,1,2,3,\ldots\} \]1.
\(n \in \mathbb{N}\)
\(P(x)=3x^{n-2}+4x^{\expfrac{5}{n}}+7\)
If \(P(x)\) is a polynomial what is the value of \(n\)?
2.
\(n \in \mathbb{N}\)
\(P(x)=2x^{\expfrac{12}{n}}+5\)
If \(P(x)\) is a polynomial what is the value of \(n\)?
3.
\(n \in \mathbb{N}\)
\(P(x)=4x^{n-3}+2x^{7-n}+9\)
If \(P(x)\) is a polynomial what is the value of \(n\)?
4.
\(n \in \mathbb{N}\)
\(P(x)=4\cdot x^{\expfrac{3n+24}{n}}\)
If \(P(x)\) is a polynomial, how many different values could ‘n’ take?