Question

How do you factor \[{x^2} - 2xy + {y^2}\]?

Answer 1

Hint: Here, we will find the factors of the product of first and third term, such that the sum of the factors is equal to the middle term. We will split the middle term into two terms and factorize the given polynomial to find the required answer.

Complete step-by-step answer:
First, we will find the product of the first term and the third term.
The first term of the given polynomial is \[{x^2}\], and the third term is \[{y^2}\].
Therefore, we get the product as \[{x^2}{y^2}\].
Now, we need to find two factors of the product \[{x^2}{y^2}\] such that their product is \[{x^2}{y^2}\], and their sum is equal to second term, that is \[ - 2xy\].
The sum of the factors \[ - xy\] and \[ - xy\] of \[{x^2}{y^2}\] is equal to the second term, that is \[ - 2xy\].
Rewriting the expression \[ - 2xy\], we get
\[ - 2xy = - xy - xy\]
Substituting \[ - 2xy = - xy - xy\] in the polynomial, we get
\[{x^2} - 2xy + {y^2} = {x^2} - xy - xy + {y^2}\]
Rewriting the terms in the expression, we get
\[ \Rightarrow {x^2} - 2xy + {y^2} = x \cdot x - x \cdot y - y \cdot x + y \cdot y\]
Factoring the terms in the expression using the distributive law of multiplication \[a \cdot b + a \cdot c = a\left( {b + c} \right)\], we get
\[ \Rightarrow {x^2} - 2xy + {y^2} = x\left( {x - y} \right) - y\left( {x - y} \right)\]
Factoring the expression again using the distributive law of multiplication \[b \cdot a + c \cdot a = \left( {b + c} \right)a\], we get
\[ \Rightarrow {x^2} - 2xy + {y^2} = \left( {x - y} \right)\left( {x - y} \right)\]
The product of a number by itself can be written as the square of a number, that is \[a \times a = {a^2}\].
Therefore, we get
\[ \Rightarrow \left( {x - y} \right)\left( {x - y} \right) = {\left( {x - y} \right)^2}\]
Thus, we have factored \[{x^2} - 2xy + {y^2}\] as \[{\left( {x - y} \right)^2}\].

Note: We have used the distributive law of multiplication in the solution to factorize the expression. The distributive law of multiplication states that \[a \cdot b + a \cdot c = a\left( {b + c} \right)\].
The given polynomial can also be factored using the algebraic identity for difference of the sum of two numbers. The square of the sum of two numbers \[a\] and \[b\] is given by the algebraic identity \[{\left( {a - b} \right)^2} = {a^2} - 2ab + {b^2}\].
The given polynomial can be factorized as
\[ \Rightarrow {x^2} - 2\left( x \right)\left( y \right) + {y^2} = {\left( {x - y} \right)^2}\]
Thus, we have factored \[{x^2} - 2xy + {y^2}\] as \[{\left( {x - y} \right)^2}\].