Question

The degree of polynomial \[\left( x+1 \right)\left( {{x}^{2}}-x-{{x}^{4}}+1 \right)\] is;
A.5
B.4
C.1
D.3

Answer 1

Hint: In the given question, we have been asked to find out the degree of the given polynomial. In order to find out the degree of the polynomial, first we need to simplify the given polynomial using the distributive property of multiplication. Then combining the like terms in the polynomial and rewrite in the form of the decreasing power. The degree of the polynomial is the greatest power of the given variable in the polynomial. Then observing the greatest power of the variable, we will get the required answer.

Complete step-by-step answer:
We have given that,
 \[\Rightarrow \left( x+1 \right)\left( {{x}^{2}}-x-{{x}^{4}}+1 \right)\]
Simplifying the above polynomial using the distributive property of multiplication;
 \[\Rightarrow x\left( {{x}^{2}} \right)+x\left( -x \right)+x\left( -{{x}^{4}} \right)+x\left( 1 \right)+1\left( {{x}^{2}} \right)+1\left( -x \right)+1\left( -{{x}^{4}} \right)+1\left( 1 \right)\]
Solving the brackets in the above polynomial, we will get
 \[\Rightarrow {{x}^{3}}-{{x}^{2}}-{{x}^{5}}+x+{{x}^{2}}-x-{{x}^{4}}+1\]
Combining the like terms, we will obtain
 \[\Rightarrow {{x}^{3}}-{{x}^{5}}-{{x}^{4}}+1\]
Rewrite the above polynomial in the form of decreasing power, we will get
 \[\Rightarrow -{{x}^{5}}-{{x}^{4}}+{{x}^{3}}+1\]
Here,
We can observe that the above polynomial has the term with the highest power of 5.
The highest power of the variable ‘x’ is 5.
Thus,
The degree of the polynomial \[\left( x-1 \right)\left( {{x}^{2}}-x-{{x}^{4}}+1 \right)\] is \[5\] .
So, the correct answer is “Option A”.

Note: While finding out the degree of the polynomial, students should need to know about the concept of the degree of the polynomial. The degree of the polynomial is the greatest power of the given variable in the polynomial. Students also need to observe the coefficient of the greatest power of variables because the coefficient of the greatest power of variables should be non-zero.