Question

How do you combine $\left( {{b}^{3}}+2{{b}^{2}}+3b \right)+\left( 4{{b}^{3}}-5{{b}^{2}}+4b \right)$?

Answer 1

Hint: To solve this question we will use a combination of the numbers of the same variable. The combinations that we will put it through will be adding up 1 and 4 together because these two are the coefficients of ${{b}^{3}}$. Similarly we will add 2 and (- 5) together and 3b and 4b with each other. After adding them together we get the required answer.

Complete step by step solution:
In the given term $\left( {{b}^{3}}+2{{b}^{2}}+3b \right)+\left( 4{{b}^{3}}-5{{b}^{2}}+4b \right)$ we can see that there is only one variable which is b. It seems to be three variables due to its powers.
We will consider $\left( {{b}^{3}}+2{{b}^{2}}+3b \right)+\left( 4{{b}^{3}}-5{{b}^{2}}+4b \right)$. First we will open the brackets to get ${{b}^{3}}+2{{b}^{2}}+3b+4{{b}^{3}}-5{{b}^{2}}+4b$. We will combine these two with the best combination. Since, there is only one combination that we can apply here and that is the following one.
For this we need to add up coefficients of like variables. Here we will add 1 and 4 together only because these two are the coefficients of ${{b}^{3}}$. Similarly we will add 2 and (- 5) together and 3b and 4b with each other. We can clearly see that 2 + (- 5) results into – 3. The minus sign will come due to the fact that it is with 5 also. And since 5 is greater than 2 so, the result will also carry the sign of 5 only. Thus, we get $\left( 1+4 \right){{b}^{3}}+\left( 2-5 \right){{b}^{2}}+\left( 3+4 \right)b=5{{b}^{3}}-3{{b}^{2}}+7b$.

Note: The combination of coefficients of like variables is best here. One cannot add the coefficient of ${{b}^{3}}$ with ${{b}^{2}}$. This means that in ${{b}^{3}}+2{{b}^{2}}+3b+4{{b}^{3}}-5{{b}^{2}}+4b$ we cannot write ${{b}^{3}}+2{{b}^{2}}=3{{b}^{3}}\,\,or,\,\,3{{b}^{2}}$. Since, this is a wrong method therefore, we have not applied it in the solution as well. If we get questions similar to this type then also, we will solve them with the same procedure as we did here. The other concept that we used here is addition and subtraction. The main focus here was to add up 2 and – 5 together. Since, there is a minus sign with one of them so we subtracted these two. Moreover, 5 being the larger number carries minus sign resulting into – 3 instead of + 3.