Question

Expand \[xy\left( {x - y} \right) + yz\left( {y - z} \right) + zx\left( {z - x} \right)\].

Answer 1

Hint: Here, we will use the distributive law of multiplication to find the products in the expression, and add them to find the expansion of \[xy\left( {x - y} \right) + yz\left( {y - z} \right) + zx\left( {z - x} \right)\]. The distributive law of multiplication states that \[a\left( {b + c} \right) = a \cdot b + a \cdot c\].

Complete step-by-step answer:
We will use the distributive law of multiplication to find the product of the numbers \[xy\] and \[\left( {x - y} \right)\], the numbers \[yz\] and \[\left( {y - z} \right)\], and the numbers \[zx\] and \[\left( {z - x} \right)\].
First, we will calculate the product of \[xy\] and \[\left( {x - y} \right)\].
The distributive law of multiplication states that \[a\left( {b + c} \right) = a \cdot b + a \cdot c\].
Substituting \[a = xy\], \[b = x\], and \[c = - y\] in the formula, we get
\[ \Rightarrow xy\left[ {x + \left( { - y} \right)} \right] = xy \cdot x + xy \cdot \left( { - y} \right)\]
Multiplying the terms and simplifying the expression, we get
\[ \Rightarrow xy\left( {x - y} \right) = {x^2}y - x{y^2}\]
Next, we will calculate the product of \[yz\] and \[\left( {y - z} \right)\].
Substituting \[a = yz\], \[b = y\], and \[c = - z\] in the formula \[a\left( {b + c} \right) = a \cdot b + a \cdot c\], we get
\[ \Rightarrow yz\left[ {y + \left( { - z} \right)} \right] = yz \cdot y + yz \cdot \left( { - z} \right)\]
Multiplying the terms and simplifying the expression, we get
\[ \Rightarrow yz\left( {y - z} \right) = {y^2}z - y{z^2}\]
Next, we will calculate the product of \[zx\] and \[\left( {z - x} \right)\].
Substituting \[a = zx\], \[b = z\], and \[c = - x\] in the formula \[a\left( {b + c} \right) = a \cdot b + a \cdot c\], we get
\[ \Rightarrow zx\left[ {z + \left( { - x} \right)} \right] = zx \cdot z + zx \cdot \left( { - x} \right)\]
Multiplying the terms and simplifying the expression, we get
\[ \Rightarrow zx\left( {z - x} \right) = {z^2}x - z{x^2}\]
Now, we will add the product of the numbers \[xy\] and \[\left( {x - y} \right)\], the numbers \[yz\] and \[\left( {y - z} \right)\], and the numbers \[zx\] and \[\left( {z - x} \right)\].
Therefore, we get
\[ \Rightarrow xy\left( {x - y} \right) + yz\left( {y - z} \right) + zx\left( {z - x} \right) = {x^2}y - x{y^2} + {y^2}z - y{z^2} + {z^2}x - z{x^2}\]
Rewriting the expression, we get
\[xy\left( {x - y} \right) + yz\left( {y - z} \right) + zx\left( {z - x} \right) = {x^2}y + {y^2}z + {z^2}x - x{y^2} - y{z^2} - z{x^2}\]
Since there are no like terms in the expression \[{x^2}y + {y^2}z + {z^2}x - x{y^2} - y{z^2} - z{x^2}\], we cannot simplify it further.
\[\therefore \] The expansion of \[xy\left( {x - y} \right) + yz\left( {y - z} \right) + zx\left( {z - x} \right)\] is \[{x^2}y + {y^2}z + {z^2}x - x{y^2} - y{z^2} - z{x^2}\].

Note: We cannot simplify \[{x^2}y + {y^2}z + {z^2}x - x{y^2} - y{z^2} - z{x^2}\] further because there are no like terms in the expression. Like terms are the terms whose variables and their exponents are the same. For example, \[100x,150x,240x,600x\] all have the variable \[x\] raised to the exponent 1. Terms which are not like, and have different variables, or different degrees of variables cannot be added together. For example, it is not possible to add \[{x^2}y\] to \[{y^2}z\] or \[ - z{x^2}\].