Question

From the sum of ${z^3} + 3{z^2} + 5z + 8$ and $4{z^3} + 2{z^2} - 7z - 2$ subtract $2{z^3} - 3{z^2} + z - 4$.
A) $0$
B) $8{z^2} - 3z$
C) $3{z^2} + 8{z^2} - 3z + 10$
D) $2 - z$

Answer 1

Hint: In order to add or subtract the equations, first we need to collect the variables with the same degrees together. Because an operation of addition or subtraction can occur only in variables of the same degrees. Therefore, collect the same degree variables from both the equations given, add their coefficients, then do the same for the subtraction part also.

Complete step by step solution:
We are given with two equations to add that are ${z^3} + 3{z^2} + 5z + 8$ and $4{z^3} + 2{z^2} - 7z - 2$, then one equation $2{z^3} - 3{z^2} + z - 4$ to subtract from the resultant of the equation add added above.
Starting with the addition part, we are adding the two equations ${z^3} + 3{z^2} + 5z + 8$ and $4{z^3} + 2{z^2} - 7z - 2$:
$\left( {{z^3} + 3{z^2} + 5z + 8} \right) + \left( {4{z^3} + 2{z^2} - 7z - 2} \right)$
To add the equations opening the parenthesis and writing the coefficients of the variables of same degree in one separate parenthesis, for example ${z^3},4{z^3}$ have common degree $3$, so writing their coefficients together:
$
  \left( {{z^3} + 3{z^2} + 5z + 8} \right) + \left( {4{z^3} + 2{z^2} - 7z - 2} \right) \\
  \left( {1 + 4} \right){z^3} + \left( {3 + 2} \right){z^2} + \left( {5 - 7} \right)z + \left( {8 - 2} \right) \\
 $
Simply solving the parenthesis and we get:
$
  \left( {1 + 4} \right){z^3} + \left( {3 + 2} \right){z^2} + \left( {5 - 7} \right)z + \left( {8 - 2} \right) \\
  5{z^3} + 5{z^2} - 2z + 6 \\
 $
Which is our obtained equation.
Therefore, the sum of ${z^3} + 3{z^2} + 5z + 8$ and $4{z^3} + 2{z^2} - 7z - 2$ is $5{z^3} + 5{z^2} - 2z + 6$.
Similarly, we need to subtract $2{z^3} - 3{z^2} + z - 4$ from the sum of the above two equations.
So, subtracting $2{z^3} - 3{z^2} + z - 4$, from the sum of ${z^3} + 3{z^2} + 5z + 8$and $4{z^3} + 2{z^2} - 7z - 2$:
$
  \left( {5{z^3} + 5{z^2} - 2z + 6} \right) - \left( {2{z^3} - 3{z^2} + z - 4} \right) \\
   \Rightarrow 5{z^3} + 5{z^2} - 2z + 6 - 2{z^3} + 3{z^2} - z + 4 \\
 $
Taking the coefficients of the variables of same degrees in parenthesis and solving it further, as we did for addition:
$
  5{z^3} + 5{z^2} - 2z + 6 - 2{z^3} + 3{z^2} - z + 4 \\
   \Rightarrow \left( {5 - 2} \right){z^3} + \left( {5 + 3} \right){z^2} + \left( { - 2 - 1} \right)z + \left( {6 + 4} \right) \\
 $
Solving the values inside the parenthesis and we get:
$
  \left( {5 - 2} \right){z^3} + \left( {5 + 3} \right){z^2} + \left( { - 2 - 1} \right)z + \left( {6 + 4} \right) \\
   \Rightarrow 3{z^3} + 8{z^2} + \left( { - 3} \right)z + 10 \\
   \Rightarrow 3{z^3} + 8{z^2} - 3z + 10 \\
 $
And, this is the resultant after subtracting $2{z^3} - 3{z^2} + z - 4$ from the sum of ${z^3} + 3{z^2} + 5z + 8$, which matches with the option C.

Therefore, the Option (C) i.e, $3{z^3} + 8{z^2} - 3z + 10$ is correct.

Note:
> Do not make a mistake by adding or subtracting the same variables with different degrees, it may result in error.
> But, the same variables with different degrees can be multiplied or divided by using the law of radicals.