Question

One factor of ${{a}^{2}}-2ab-{{c}^{2}}+{{b}^{2}}$ is _____.

Answer 1

Hint: We will be using the concepts of factorization to solve the problem. We will start by using the identity \[{{\left( A-B \right)}^{2}}={{A}^{2}}+{{B}^{2}}-2AB\] to simplify the given expression into ${{\left( a-b \right)}^{2}}-{{c}^{2}}$. Then we will use the identity ${{A}^{2}}-{{B}^{2}}=\left( A-B \right)\left( A+B \right)$to factorize it and find the final answer. Now, we have to find a factor of ${{a}^{2}}-2ab-{{c}^{2}}+{{b}^{2}}$. For this we have to factorize and then we will be able to get its factor.

Complete step-by-step answer:
Let us get started with our solution. So, first we will consider the equation that has been given to us as
${{a}^{2}}-2ab-{{c}^{2}}+{{b}^{2}}$
If we notice carefully, we have terms in a and b such that they are in the form of an identity.
Now, we know the identity that
${{\left( A-B \right)}^{2}}={{A}^{2}}+{{B}^{2}}-2AB$
Therefore, we have,
${{a}^{2}}-2ab-{{c}^{2}}+{{b}^{2}}={{\left( a-b \right)}^{2}}-{{c}^{2}}$
The above expression has been reduced such that it represents the difference of squares of two terms.
Now, we have an identity given by
${{A}^{2}}-{{B}^{2}}=\left( A-B \right)\left( A+B \right)$
Therefore, applying this, we have,
${{a}^{2}}-2ab-{{c}^{2}}+{{b}^{2}}=\left( a-b-c \right)\left( a-b+c \right)$
Now, this has been reduced to the extent where we have two factors.
So, finally we have,
${{a}^{2}}-2ab-{{c}^{2}}+{{b}^{2}}=\left( a-b-c \right)\left( a-b+c \right)$
Now, the factors of ${{a}^{2}}-2ab-{{c}^{2}}+{{b}^{2}}$ are $\left( a-b-c \right)\ and\ \left( a-b+c \right)$. Therefore, the answer can be either $\left( a-b-c \right)\ or\ \left( a-b+c \right)$. Since, we are required to find one factor.

Note: It is important to remember basic algebraic identities to solve the problem like,
\[{{\left( A+B \right)}^{2}}={{A}^{2}}+{{B}^{2}}-2AB\]
\[{{\left( A-B \right)}^{2}}={{A}^{2}}+{{B}^{2}}-2AB\]
\[{{A}^{2}}-{{B}^{2}}=\left( A-B \right)\left( A+B \right)\]
It is also important to note that the answer can be either $\left( a-b-c \right)\ or\left( a-b+c \right)$. Since we are required to find only one factor of ${{a}^{2}}+{{b}^{2}}-2ab-{{c}^{2}}$.
Students must not try to club the terms of a and c and try to apply the identity ${{A}^{2}}-{{B}^{2}}=\left( A-B \right)\left( A+B \right)$. This might get confusing and take time to simplify and obtain the factors.