Question

How do you multiply out \[{\left( {5{z^2} + 3z + 2} \right)^2}\]?

Answer 1

Hint: We use the square of the given value and open it in terms of multiplication with itself. Use distributive property to multiply each term from the first bracket with each term of second bracket separately.
* Distributive Property: For any three numbers ‘a’, ‘b’ and ‘c’ we can write \[a(b + c) = ab + bc\]

Complete step-by-step solution:
We have to multiply out the square of quadratic equation i.e. \[{\left( {5{z^2} + 3z + 2} \right)^2}\]
We know square means the number or the value is multiplied to itself one time i.e. \[{a^2} = a \times a\]
We can write \[{\left( {5{z^2} + 3z + 2} \right)^2} = \left( {5{z^2} + 3z + 2} \right) \times \left( {5{z^2} + 3z + 2} \right)\]
Now we use distributive property to break the product of brackets
\[ \Rightarrow \left( {5{z^2} + 3z + 2} \right) \times \left( {5{z^2} + 3z + 2} \right) = 5{z^2}\left( {5{z^2} + 3z + 2} \right) + 3z\left( {5{z^2} + 3z + 2} \right) + 2\left( {5{z^2} + 3z + 2} \right)\]
Now we again use distributive property for each term in right hand side of the equation
\[ \Rightarrow \left( {5{z^2} + 3z + 2} \right) \times \left( {5{z^2} + 3z + 2} \right) = 5{z^2} \times 5{z^2} + 5{z^2} \times 3z + 5{z^2} \times 2 + 3z \times 5{z^2} + 3z \times 3z + 3z \times 2 + 2 \times 5{z^2} + 2 \times 3z + 2 \times 2\]Calculate each of the products on right hand side of the equation.
Use law of exponents i.e. add the powers when base is same
\[ \Rightarrow \left( {5{z^2} + 3z + 2} \right) \times \left( {5{z^2} + 3z + 2} \right) = 25{z^{2 + 2}} + 15{z^{2 + 1}} + 10{z^2} + 15{z^{2 + 1}} + 9{z^{1 + 1}} + 6z + 10{z^2} + 6z + 4\]
Add the values in the power for each term in right hand side of the equation
\[ \Rightarrow \left( {5{z^2} + 3z + 2} \right) \times \left( {5{z^2} + 3z + 2} \right) = 25{z^4} + 15{z^3} + 10{z^2} + 15{z^3} + 9{z^2} + 6z + 10{z^2} + 6z + 4\]
Now we combine the terms having same variable associated with it
\[ \Rightarrow \left( {5{z^2} + 3z + 2} \right) \times \left( {5{z^2} + 3z + 2} \right) = 25{z^4} + (15{z^3} + 15{z^3}) + (10{z^2} + 9{z^2} + 10{z^2}) + (6z + 6z) + 4\]
Add the terms in respective brackets
\[ \Rightarrow \left( {5{z^2} + 3z + 2} \right) \times \left( {5{z^2} + 3z + 2} \right) = 25{z^4} + 30{z^3} + 29{z^2} + 12z + 4\]

\[\therefore \]The value of \[{\left( {5{z^2} + 3z + 2} \right)^2}\] is \[25{z^4} + 30{z^3} + 29{z^2} + 12z + 4\].

Note: Many students make the mistake of solving this question by writing the quadratic equation in factored form as it is clearly visible that it can be written in the form of product of its factors using factorization method. They end the answer with powers of factors which are wrong. Keep in mind we have to multiply out, this means we have to calculate the final value which will be obtained after multiplying all the values together i.e. opening the brackets and squares to its maximum.