Question

Write the index form of the polynomial given below using variable $x$ from its coefficient from $\left( 5,0,0,0,0,-7 \right)$.

Answer 1

Hint: In this problem we need to write the index form of the polynomial by using the variable $x$ with the given coefficients. For this we need first look at the given coefficients and write the number of coefficients given. The given last coefficient represents the constant term, last second coefficients represent the coefficients of the variable ${{x}^{1}}$ and the third coefficient from last represents the coefficient of the variable ${{x}^{2}}$. Likewise we will write what the given coefficient represents. After we will sum all the variables which are represented by the coefficients to get the required polynomial.

Complete step by step solution:
Given coefficients are $\left( 5,0,0,0,0,-7 \right)$.
Using the variable $x$.
The last coefficient which is $-7$ represents the constant. Hence constant is $-7$.
The second last coefficient which is $0$ represents the coefficient of the variable ${{x}^{1}}=x$.
The second last coefficient which is $0$ represents the coefficient of the variable ${{x}^{2}}$.
The second last coefficient which is $0$ represents the coefficient of the variable ${{x}^{3}}$.
The second last coefficient which is $0$ represents the coefficient of the variable ${{x}^{4}}$.
The second last coefficient which is $5$ represents the coefficient of the variable ${{x}^{5}}$.
Hence the polynomial is $5{{x}^{5}}+0{{x}^{4}}+0{{x}^{3}}+0{{x}^{2}}+0{{x}^{1}}-7$
When we multiply a variable with zero, then we will get zero as a result. Hence the above equation is modified as $5{{x}^{5}}-7$.

Note: We can use alternate methods to solve this problem. We can directly assume the polynomial based on the number of coefficients. If $n$ is the number of coefficients given then assume the $a{{x}^{n-1}}+b{{x}^{n-2}}+c{{x}^{n-3}}+.....+{{x}_{0}}$ where $\left( a,b,c,...,{{x}_{0}} \right)$ are given set of coefficients.