Question

Solve $4y\left( {3{y^2} + 5y - 7} \right) + 2\left( {{y^3} - 4{y^2} + 5} \right)$.

Answer 1

Hint: First expand the whole function and then simplify it so that you can form additive and subtractive connections between the terms which shall be the accepted answer. Also remember, this is a function and not an equation because it is not equal to any finite value.

Complete step by step solution:
In this question, the first course of action will be to expand the function because if we can’t expand it, we will be unable to simplify it. Also, for the sake of simplicity, let us assume a name for this function, say $f\left( y \right)$.
So now $f\left( y \right)$ can be given by,
\[f\left( y \right) = 4y\left( {3{y^2} + 5y - 7} \right) + 2\left( {{y^3} - 4{y^2} + 5} \right)\]
Moving forward, we multiply the coefficient of the first subpart with the first subpart and the coefficient of the second subpart with the second subpart, arrange the in order of their power and then mathematically simplify the formed equation.
\[\
  f\left( y \right) = 4y\left( {3{y^2} + 5y - 7} \right) + 2\left( {{y^3} - 4{y^2} + 5} \right) \\
   = \left[ {4y \times 3{y^2} + 4y \times 5y - 4y \times 7} \right] + \left[ {2 \times {y^3} - 2 \times 4{y^2} + 2 \times 5} \right] \\
   = 12{y^3} + 20{y^2} - 28y + 2{y^3} - 8{y^2} + 10 \\
   = 12{y^3} + 2{y^3} + 20{y^2} - 8{y^2} - 28y + 10 \\
   = 14{y^3} + 12{y^2} - 28y + 10 \\
\ \]
Here, what we have done is multiplied \[4y\] with every member of the box which it is coefficient of and then multiplied 2 with every member of the group that it is coefficient of.
This way, we reached the third step but then there are similar objects in the function subject to addition and subtraction so we simplified everything by arranging them in descending order of their powers as shown in the fourth step. Finally, we simplify everything to get our answer in the fifth step.

Note:
Since the function was not given to be equal to zero, we were not obliged to find its roots and so just simplifying it shall give us the acceptable answer but it is always advised that if possible, take the function into multiplicative form, although the same is not possible in this question.