Question

How do you combine like terms in $ \left( {6{n^3} - 7 + 6{n^2}} \right) + \left( {4 - 7{n^2} - 6{n^4}} \right) + \left( {8{n^4} - 8n} \right) $ ?

Answer 1

Hint: We know that the like terms can be defined as the terms whose variables as well as their exponents are the same. Therefore, here, we will first open the brackets. After that we will find the like terms and perform the mathematical operation addition or subtraction accordingly.

Complete step-by-step answer:
We are given the polynomial with variable \[n\] :
 $ \left( {6{n^3} - 7 + 6{n^2}} \right) + \left( {4 - 7{n^2} - 6{n^4}} \right) + \left( {8{n^4} - 8n} \right) $
Now, let us first open the bracket so that we will get:
 $ \Rightarrow 6{n^3} - 7 + 6{n^2} + 4 - 7{n^2} - 6{n^4} + 8{n^4} - 8n $
Now, we need to find the like terms in order to combine them.
Here, we can see that the highest degree of variable \[n\]is 4 and there are two terms with variable \[n\]and its exponent 4 which are $ 6{n^4} $ and $ 8{n^4} $ . Thus, these two can be combined as they are like terms.
Moreover, There are two terms with variable \[n\]and its exponent 2 which are $ 6{n^2} $ and $ 7{n^2} $ . Hence, these two can also be combined as they are like terms.
Now, we can see that there is only a single term with power 3 and one of the variables \[n\] in the polynomial. However, there are two constant terms or we can say that the terms with the variable n and its exponent zero which are 7 and 4. These two are also considered as like terms and can be combined
Now we will write the like terms together before combining them.
\[
   \Rightarrow - 6{n^4} + 8{n^4} + 6{n^3} + 6{n^2} - 7{n^2} - 8n - 7 + 4 \\
   \Rightarrow \left( { - 6 + 8} \right){n^4} + 6{n^3} + \left( {6 - 7} \right){n^2} - 8n - 3 \\
   \Rightarrow 2{n^4} + 6{n^3} - {n^2} - 8n - 3 \;
 \]
Thus, this is how we can combine like terms in any polynomial.
So, the correct answer is “ $ 2{n^4} + 6{n^3} - {n^2} - 8n - 3 $ ”.

Note: As we have seen, like terms can be identifier depending on only a variable's power. The coefficients of these variables can be different. We are ultimately using these coefficients of like terms when we combine them as we have done in this problem.