Question

If the polynomial p(y) is defined as $p\left( y \right)=2+y+{{y}^{2}}-{{y}^{3}}$ then find the value of p(0) p(1) and p(2).

Answer 1

Hint: Now we are given with that the polynomial p(y) is defined as $p\left( y \right)=2+y+{{y}^{2}}-{{y}^{3}}$. This means for each y = a the value of polynomial is $p\left( y \right)=2+a+{{a}^{2}}-{{a}^{3}}$ . Hence to find the values p(0) p(1) and p(2) we will substitute y = 0, 1 and 2 in the equation respectively.

Complete step-by-step answer:
Now let us first understand the term polynomial.
A polynomial is an expression which includes variables, exponents and coefficient. Though the exponents must always be whole numbers. There can be one or more than one variables in a polynomial.
Consider the given polynomial.
$p\left( y \right)=2+y+{{y}^{2}}-{{y}^{3}}$
The given polynomial is a polynomial in one variable which is y.
To find the value of p(0) we will substitute y = 0 in the equation. Hence we get
$p\left( 0 \right)=2+0+{{0}^{2}}-{{0}^{3}}$
Hence the value of p(0) = 2.
Now similarly let us find the value of p(1) by substituting y = 1.
Hence we get, $p\left( 1 \right)=2+1+{{1}^{2}}-{{1}^{3}}=2+1+1-1=3$
Hence we have p(1) = 3.
Hence the value of p(1) = 3
Now again if we substitute y = 2 we will get the value of p(2).
$p\left( 2 \right)=2+2+{{2}^{2}}-{{2}^{3}}=2+2+4-8=0$
Hence we get p(2) = 0.
Hence the value of p(2) = 0.

Note: Now note that since we get p(2) = 0 then 2 is called the root of the polynomial. Root means the value at which the polynomial becomes 0. Also note that the polynomial need not always have a root. For example consider a constant polynomial, y = 2. Since the polynomial has a constant value of 2 it will never be 0 and hence the polynomial has no root.