Question

How will you find the factor of the expression \[5{{x}^{9}}+10{{x}^{8}}-15{{x}^{7}}+10{{x}^{^{6}}}\]?

Answer 1

Hint: In the above question that is mentioned for finding the factor of the equation we will have to find the highest common factor of the whole equation i.e. we need to find the highest term which will divide every single term and give its lowest value.

Complete step-by-step answer:
The expression that is mentioned is \[5{{x}^{9}}+10{{x}^{8}}-15{{x}^{7}}+10{{x}^{^{6}}}\]. In this expression we will first be observing which of the things are common in the whole term. For this when we go to the mentioned equation we can clearly see that the coefficient of the different terms is in multiple of 5, so we can say that we will have to take the highest multiple of 5 which will leave the remainder as 1 from all the terms like we take the first term we can see that 5 will divide the coefficient and will give the remainder as one. We can also see that the 2nd and 4th term of the equation has equal coefficient with highest common factor as 10, but we saw that the 1st term of the equation will have the highest common factor as 5 so 10 will not come in the highest common factor, and the same with the 3rd term of the equation.
So from the coefficient of the equation the factor would be equal to 5.
Now for the x term of the equation, for this also we will use the same method and we will get that \[{{x}^{6}}\] from the 4th term is the highest common factor for all terms in the equation.
Now we can finally write the above equation as:
 \[\Rightarrow 5{{x}^{9}}+10{{x}^{8}}-15{{x}^{7}}+10{{x}^{^{6}}}=5{{x}^{6}}\left( {{x}^{3}}+2{{x}^{2}}-3x+2 \right)\]
From the above equation we can say that \[5{{x}^{6}}\] is the factor of the given expression and hence \[5{{x}^{6}}\]divides all the terms of the equation.

Note: Always remember the definition of what the question is asking us to find, like in this question we had to find out the factor of an expression and by definition of factor we had to find the highest common factor of the equation which will divide the whole term in numerical form and not in any other kind of form, as in the above question \[5{{x}^{6}}\] divides each and every single term by giving its lowest numerical value and doesn’t change it to fraction form.