Question

Write the following in expanded form:
 $ {(2x - y + z)^2} $

Answer 1

Hint: Expanded form a mathematical notation is the expansion of the given term in a simpler form that is they form a mathematical expression in which numbers are separated into individual place values. It can be done in several ways but in the given question, we have to expand the square of the term to get the math value of the individual digits. Many identities can be used to solve the square of two or three numbers of terms. Using the appropriate identity, we can find the answer to the given question.

Complete step-by-step answer:
We know that the square of the sum of three terms is equal to the sum of the square of the first term, the square of the second term, square of the third term, twice the product of first two terms, twice the product of last two terms, and twice the product of first and third term. That is -
 $ {(a + b + c)^2} = {a^2} + {b^2} + {c^2} + 2ab + 2bc + 2ac $
In the given question, $ a = 2x,\,b = - y\,and\,c = z $
So,
 $
\Rightarrow {(2x - y + z)^2} = {(2x)^2} + {( - y)^2} + {(z)^2} + 2 \times 2x \times ( - y) + 2 \times ( - y) \times z + 2 \times 2x \times z \\
\Rightarrow {(2x - y + z)^2} = 4{x^2} + {y^2} + {z^2} - 4xy - 2yz + 4xz \;
  $
Thus the expanded form of $ {(2x - y + z)^2} $ is $ 4{x^2} + {y^2} + {z^2} - 4xy - 2yz + 4xz $ .
So, the correct answer is “ $ 4{x^2} + {y^2} + {z^2} - 4xy - 2yz + 4xz $ ”.

Note: There are many identities for finding the square and cube of two and three terms. In the above question, we have three terms and the power is two, so the appropriate formula to be used is $ {(a + b + c)^2} = {a^2} + {b^2} + {c^2} + 2ab + 2bc + 2ac $ .
Instead of applying the identity, we can also write $ {(2x - y + z)^2} = (2x - y + z)(2x - y + z) $ , Multiplying these two with each other, we can obtain the same answer.