Question

If we are given a polynomial function as \[f\left( x \right)=2{{x}^{3}}+3{{x}^{2}}-11x+6\] , then find the value of \[f\left( -1 \right)\] .

Answer 1

Hint: The given polynomial is \[f\left( x \right)=2{{x}^{3}}+3{{x}^{2}}-11x+6\] . Compare \[f\left( x \right)\] and \[f\left( -1 \right)\] to get the value of x. Now, put that value of x in the polynomial \[f\left( x \right)=2{{x}^{3}}+3{{x}^{2}}-11x+6\] to get the value of \[f\left( -1 \right)\] .

Complete step-by-step solution
According to the question, we are given a polynomial \[f\left( x \right)\] and we have to find the value of \[f\left( -1 \right)\] .
In mathematics, we know that the polynomial is an expression that consists of variables, coefficients, constants, and exponents that involves addition, subtraction, multiplication, and division operations.
The given polynomial = \[f\left( x \right)=2{{x}^{3}}+3{{x}^{2}}-11x+6\] ………………………………………(1)
In the above equation, we can see that the given polynomial has a variable in terms of x.
We are asked to find the value of \[f\left( -1 \right)\] for the given polynomial, \[f\left( x \right)=2{{x}^{3}}+3{{x}^{2}}-11x+6\] .
Let us compare \[f\left( x \right)\] and \[f\left( -1 \right)\] .
On comparison, we can say that when \[x\] is replaced by -1 in \[f\left( x \right)\] , then \[f\left( -1 \right)\] is obtained.
Now, using the above logic and on putting \[x=-1\] in equation (1), we get
\[\Rightarrow f\left( x \right)=2{{x}^{3}}+3{{x}^{2}}-11x+6\]
\[\Rightarrow f\left( -1 \right)=2{{\left( -1 \right)}^{3}}+3{{\left( -1 \right)}^{2}}-11\left( -1 \right)+6\] ………………………………………………(2)
In the above equation, we can observe that it needs to be solved and simplified further to obtain an integral value.
Solving, \[{{\left( -1 \right)}^{3}}=\left( -1 \right)\left( -1 \right)\left( -1 \right)=-1\] ……………………………………..(3)
Similarly, solving, \[{{\left( -1 \right)}^{2}}=\left( -1 \right)\left( -1 \right)=1\] …………………………………………………(4)
Now, using equation (3) and equation (4), and on substituting \[{{\left( -1 \right)}^{3}}\] by -1 and \[{{\left( -1 \right)}^{2}}\] by 1 in equation (2), we get
\[\begin{align}
  & \Rightarrow f\left( -1 \right)=2\left( -1 \right)+3\left( 1 \right)-11\left( -1 \right)+6 \\
 & \Rightarrow f\left( -1 \right)=-2+3+11+6 \\
\end{align}\]
\[\Rightarrow f\left( -1 \right)=18\] ………………………………………..(5)
From equation (5), we have got the value of \[f\left( -1 \right)\] that is the value of \[f\left( -1 \right)\] is 18.
Therefore, the value of \[f\left( -1 \right)\] is 18.

Note: Whenever this type of question appears where we are given a polynomial \[f\left( x \right)\] and asked the value of \[f\left( a \right)\] . The best way to solve it is, just replace x by \[a\] in the polynomial \[f\left( x \right)\] and calculate the value of \[f\left( a \right)\] . Some student try to solve the given polynomial by factorization but that would be a longer approach an it would take time to find the value.