Question

what is the coefficient of ${{x}^{2}}$ in each of the following.
(a) $2+{{x}^{2}}+x$
(b) $2-{{x}^{2}}+x$
(c) $\dfrac{\pi }{2}{{x}^{2}}+x$
(d) $\sqrt{2x}-1$

Answer 1

Hint:In this question, we will use the concept of coefficient in an algebraic expression, to find the coefficient of ${{x}^{2}}$. The number or term, which is multiplied to ${{x}^{2}}$ will be its coefficient. We will check each expression and find the coefficient of ${{x}^{2}}$ .

Complete step-by-step answer:
In algebra, the coefficient of a variable in any algebraic expression is the number along with its sign, positive or negative, which is multiplied by that variable in that term. Coefficient of a variable is not necessarily a single number but can also contain other variables, but not same variables.
Example, $2xy+5$, here coefficient of x is 2y and not 2 alone as x is multiplied by 2y.
Let us now find out the coefficient of ${{x}^{2}}$ in expression given in question.
Now, the first expression given to us in the question is $2+{{x}^{2}}+x$.
Here, we have 3 terms $2,\,\,{{x}^{2}}$ and $x$. In terms containing ${{x}^{2}}$, there is no number in front of it, but 1 is a factor of all numbers. So, the coefficient of ${{x}^{2}}$ in this expression is 1.
Second expression given to us in a question is, $2-{{x}^{2}}+x$
Here, we have 3 terms $2,\,\,{{x}^{2}}$ and ${{x}^{3}}$. In terms containing ${{x}^{2}}$, there is no number multiplied by it but it has a negative sign and also 1 is a factor of all the numbers. So, the coefficient of ${{x}^{2}}$ in this expression is -1.
Third expression given to us in a quadratic equation is, $\dfrac{\pi }{2}{{x}^{2}}+x$.
Here we have 2 terms $\dfrac{\pi }{2}{{x}^{2}}$ and $x$. In terms containing ${{x}^{2}}$, ${{x}^{2}}$ is multiplied with $\dfrac{\pi }{2}$.
So, the coefficient of ${{x}^{2}}$ in this expression is $\dfrac{\pi }{2}$.
Fourth expression given to us in the question is $\sqrt{2x}-1$.
Here we have 2 terms $\sqrt{2x}$ and $-1$. See, in this expression there is no term containing ${{x}^{2}}$. But we can add 0 to this expression without changing it and 0 can be written as $0\times {{x}^{2}}$. So, the coefficient of ${{x}^{2}}$ in this expression is 0.

Note: While finding the coefficient, take note that if the variable whose coefficient is to be found, is not present, then its coefficient is taken as 0. Students get confused in such cases and are unable to answer the problem. It should be kept in mind that the coefficient of a variable which is not present in an expression is always 0.