Question

Find the coefficient of x in the following equation.
\[2{{\text{x}}^2} + {\text{x}} - {3^8}\]

Answer 1

Hint: Let us reduce the polynomial equation (i.e. put the value of \[{3^8}\]). So, we get a standard quadratic equation \[{\text{a}}{{\text{x}}^2} + {\text{bx + c}}\], and the coefficient of any term \[{{\text{x}}^n}\] is the constant term attached to it.

Complete step-by-step answer:
As we know that the value of \[{3^8}\] is a product of eight times three.
So, \[{3^8}\] = 3*3*3*3*3*3*3*3 = 6561
So, the given equation is written as \[2{{\text{x}}^2} + {\text{x}} - 6561\].
Now as we know that in any polynomial equation \[{\text{a}}{{\text{x}}^n} + {\text{b}}{{\text{x}}^{n - 1}} + {\text{c}}{{\text{x}}^{n - 2}} + {\text{d}}{{\text{x}}^{n - 3}}.... + {\text{ex}} + {\text{f}}\]. Where a, b, c, d, e and f are the constants and x are the variable. And n can have any value.
Now the coefficient of \[{{\text{x}}^t}\] (t can be any value) in this equation will be the constant term attached to it in the equation.
So, the coefficient of \[{{\text{x}}^n}\] in the above polynomial equation will be a.
Coefficient of \[{{\text{x}}^{n - 1}}\] in the above polynomial equation will be b.
Coefficient of \[{{\text{x}}^{n - 2}}\] in the above polynomial equation will be c.
Coefficient of \[{{\text{x}}^{n - 3}}\] in the above polynomial equation will be d.
Coefficient of \[{\text{x}}\] in the above polynomial equation will be e.
And the coefficient of \[{{\text{x}}^0}\] in the above polynomial equation will be f.
So, now we have to find the coefficient of the x in the given equation \[2{{\text{x}}^2} + {\text{x}} - 6561\].
We can write the above equation as \[2{{\text{x}}^2} + 1{\text{x}} - 6561\]
As we can see that constant with x is 1.
Hence, the coefficient of x in the equation \[2{{\text{x}}^2} + {\text{x}} - {3^8}\] will be 1.

Note: In such types of questions if we are asked to find the coefficient of term \[{{\text{x}}^n}\]. And \[{{\text{x}}^n}\] is not present in the equation then the quotient of \[{{\text{x}}^n}\] will be 0, because 0*\[{{\text{x}}^n}\] = 0, and if no constant term is attached with \[{{\text{x}}^n}\] then the coefficient of \[{{\text{x}}^n}\] will be 1, because 1*\[{{\text{x}}^n}\] = \[{{\text{x}}^n}\]. Otherwise the coefficient of \[{{\text{x}}^n}\] will be the constant term attached with \[{{\text{x}}^n}\].