Question

What is the coefficient of \[{x^3}\] for the polynomial \[{x^6} + 3{x^5} - 4{x^3} + 2x = 0\]?
A.$3$
B.$ - 4$
C.$1$
D.$0$

Answer 1

Hint: The coefficient of the variable is defined as the product factor of a remaining term. Or it can be defined as a specific variable coefficient in which we are treating everything else besides that variable (and its exponent) as part of the coefficient.
In this question it is given that we have to find the coefficient of \[{x^3}\]among the given polynomials. So for this we have to know that if any algebraic expression is given like this, \[{x^6} + 3{x^5} - 4{x^3} + 2x = 0\] then the coefficient of \[{x^3}\]will be the numerical or constant quantity placed before and multiplied with \[{x^3}\].

Complete step-by-step answer:
We have been given the equation \[{x^6} + 3{x^5} - 4{x^3} + 2x = 0\].
We have to find the coefficient of \[{x^3}\]in the above polynomial.
So, the above algebraic equation can be written as,
\[ \Rightarrow 1 \times {x^6} + 3 \times {x^5} - 4 \times {x^3} + 2 \times x = 0\]
Here, -4 is multiplied with \[{x^3}\].
So, the coefficient of \[{x^3}\] for the polynomial \[{x^6} + 3{x^5} - 4{x^3} + 2x = 0\] is $ - 4$.
Therefore, option (B) is the correct answer.

Note: When we are solving this type of question, you need to know that if the term that you are asked to find is negative then you have to make it positive by taking minus in the bracket and the coefficient becomes negative, like we did in this problem.
Often coefficients are numbers as in this question, although they could be two or multiple variables in the expression. In such a case one must clearly distinguish between the variable whose coefficient needed to be found as in mentioned in the below example.
Here the coefficient of \[{x^3}\] for the polynomial \[3{x^3}y + 3{y^4} + {x^2}{y^5}\;\] is $3y$. So, as we can see it is a combination of a variable as well as the constant.