Expand $\left(x-\frac{1}{3} y+\frac{1}{5} z\right)^2$
Answer (1)
Apply the formula for the square of a trinomial: ( a + b + c ) 2 = a 2 + b 2 + c 2 + 2 ab + 2 a c + 2 b c .
Substitute a = x , b = − 3 1 y , and c = 5 1 z into the formula.
Simplify the expression to obtain the final expanded form.
The expanded form is: x 2 + 9 1 y 2 + 25 1 z 2 − 3 2 x y + 5 2 x z − 15 2 yz .
Explanation
Understanding the Problem We are asked to expand the square of a trinomial: ( x − 3 1 y + 5 1 z ) 2 .
Applying the Trinomial Square Formula We will use the formula ( a + b + c ) 2 = a 2 + b 2 + c 2 + 2 ab + 2 a c + 2 b c , where a = x , b = − 3 1 y , and c = 5 1 z .
Expanding and Simplifying Substitute a = x , b = − 3 1 y , and c = 5 1 z into the formula:
( x − 3 1 y + 5 1 z ) 2 = x 2 + ( − 3 1 y ) 2 + ( 5 1 z ) 2 + 2 ( x ) ( − 3 1 y ) + 2 ( x ) ( 5 1 z ) + 2 ( − 3 1 y ) ( 5 1 z ) = x 2 + 9 1 y 2 + 25 1 z 2 − 3 2 x y + 5 2 x z − 15 2 yz
Final Result Thus, the expanded form of the given expression is:
x 2 + 9 1 y 2 + 25 1 z 2 − 3 2 x y + 5 2 x z − 15 2 yz
Examples
Expanding squared trinomials is useful in various fields, such as physics and engineering, when dealing with areas or volumes that depend on multiple variables. For example, if you are calculating the area of a rectangular garden with sides that are expressed as trinomials, you would need to expand the square of a trinomial to find the total area. Similarly, in physics, when dealing with kinetic energy involving multiple velocity components, you might encounter squared trinomials.
