Question

How do you factor $ay-yz+ax-xz$?

Answer 1

Hint: The polynomial $ay-yz+ax-xz$, which is given in the above question, consists of four terms. So we can form two groups of the terms. For this we need to combine the first two terms, $ay$ and $-yz$, to form the first group, and the last two terms, $ax$ and $-xz$, to form the second group. The combination will look like $\left( ay-yz \right)+\left( ax-xz \right)$. Then we have to take factors common from each of the two groups so that we will be left with one factor common to both of the groups. Taking the common factor outside will factor the given polynomial completely.

Complete step by step solution:
The polynomial given in the above question is
$\Rightarrow p=ay-yz+ax-xz$
Let us combine the first two and the last two terms together to get
$\Rightarrow p=\left( ay-yz \right)+\left( ax-xz \right)$
Now, we can take $y$ and $x$ common from the first and the second group respectively to get
$\Rightarrow p=y\left( a-z \right)+x\left( a-z \right)$
Finally, we can take the common factor $\left( a-z \right)$ outside to get
$\Rightarrow p=\left( a-z \right)\left( y+x \right)$
The above polynomial is the completely factored form which cannot be factored further.
Hence, we have factored the polynomial given in the above question as $\left( a-z \right)\left( y+x \right)$.

Note:
In the above solution, we have combined the first two terms, $ay$ and $-yz$, to form the first group, and the last two terms, $ax$ and $-xz$, to form the second group. You can also try this solution by combining the first term $ay$ with the third term $ax$ and similarly second term $-yz$ with the fourth term $-xz$ and proceed similarly to get the same answer which is obtained above.