Question

Check whether \[g(x)\] is a factor of \[p(x)\] by dividing polynomial by \[p(x)\] by polynomial \[g(x)\],
Where \[p(x)={{x}^{5}}-4{{x}^{3}}+{{x}^{2}}+3x+1\], \[g(x)={{x}^{3}}-3x+1\]

Answer 1

Hint: To solve this question we should first know about what is the factor and how we can determine whether the given quantity is the factor of some other quantity or not. We can determine it by taking the division and if the remainder is zero then that quantity is the factor of the given number.

Complete step-by-step solution:
To solve this question we should know that in general a factor is referred to as the number that is multiplied by some specific number and will produce the given number. From this we can say that \[g(x)\] is a factor of \[p(x)\] then it is sure that when we will divide \[p(x)\] by \[g(x)\]the remainder must be zero (\[0\]) .
Let us consider some examples to get a clear idea of this statement and then we can easily solve the given question.
We all know that \[2\] is a factor of \[20\] and when we divide \[20\] by \[2\] we will get \[10\] as a quotient whereas \[0\] as a remainder. Similarly, if we consider \[2\] it is not a factor of \[21\] we are able to say this because if we will divide \[21\] by \[2\] we will now again get \[10\] as a quotient but in this case we will get a remainder which is equal to the \[1\].
Now it is clear that if we want to check for any number or any polynomial is a factor of the given number or polynomial the remainder must be zero.
Now let us apply this concept in the given question to check whether the given polynomial \[g(x)\]is a factor of \[p(x)\] or not.
So divide the polynomial \[p(x)\] by \[g(x)\], we will get
\[\begin{align}
  & {{x}^{3}}-3x+1)\overline{{{x}^{5}}-4{{x}^{3}}+{{x}^{2}}+3x+1}({{x}^{2}}-1 \\
 &\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; {{\text{x}}^{5}}-3{{x}^{3}}+{{x}^{2}} \\
 & \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\overline{\begin{align}
  &\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; -{{x}^{3}}+3x+1 \\
 & \;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \underline{-{{x}^{3}}+3x-1} \\
 & \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\text{2}} \\
\end{align}} \\
\end{align}\]
From this division it is clear that after dividing \[p(x)\] by \[g(x)\] there is the remainder which is equal to \[2\]. So we must say that \[g(x)\] is not a factor of \[p(x)\].
Hence we can conclude that polynomial \[g(x)={{x}^{3}}-3x+1\] is not a factor of polynomial\[p(x)={{x}^{5}}-4{{x}^{3}}+{{x}^{2}}+3x+1\].

Note: Polynomial word can be divided into two words ‘Poly’ which means ‘many’ and ‘Nominal’ which means ‘terms’. Polynomials are composed of variables and coefficients. The operations which are performed between the two terms are addition, subtraction, multiplication but we cannot perform division in between them.