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Find $p\left( 0 \right)$ , $p\left( 1 \right)$ and $p\left( 2 \right)$ for $p\left( z \right) = {z^3}$
A) 0,1,8
B) 0,1,7
C) \[-1\],2,6
D) 2,5,8

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Last updated date: 20th Jun 2024
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Answer
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Hint: Here we need to find the value of the polynomial function at different values of the variable. So, we will substitute different values of the variable in the given polynomial function. Then we will use the mathematical operation to get the required value of polynomials at different values of the variables.

Complete step by step solution:
Here we need to find the value of the polynomial function at different values of the variable.
The given polynomial function is $p\left( z \right) = {z^3}$
We will first find the value of $p\left( 0 \right)$. For that, we will substitute the value of variable $z$ as zero.
On substituting the value of the variable as zero, we get
$p\left( 0 \right) = {\left( 0 \right)^3}$
On further simplification, we get
$ \Rightarrow p\left( 0 \right) = 0$
We will first find the value of $p\left( 1 \right)$. For that, we will substitute the value of variable $z$ as 1.
On substituting the value of the variable as zero, we get
$ p\left( 1 \right) = {\left( 1 \right)^3}$
On further simplification, we get
$ \Rightarrow p\left( 1 \right) = 1$
We will first find the value of $p\left( 2 \right)$. For that, we will substitute the value of variable $z$ as 2.
On substituting the value of the variable as zero, we get
$ p\left( 2 \right) = {\left( 2 \right)^3}$
On further simplification, we get
$ \Rightarrow p\left( 2 \right) = 8$
Therefore, the values of $p\left( 0 \right)$, $p\left( 1 \right)$ and $p\left( 2 \right)$ are 0, 1 and 8 respectively.

Hence, the correct option is option A.

Note:
We know that a polynomial function is defined as the function which involves only non-negative integer powers or only positive integer exponents of a variable in an equation such as the quadratic equation, cubic equation, etc. Here, the given polynomial equation is a cubic equation. A cubic equation is defined as an equation which has the highest degree of 3 and has 3 solutions. Similarly, a quadratic equation is defined as an equation which has the highest degree of 2 and has 2 solutions. So, we can say that the number of solutions depends on the highest degree of the equation.