Find: $\frac{\left({\mathrm{y}}^2-7\mathrm{y}+10\right)}{(\mathrm{y}-5)}$.

$\left(y^2-7y+10\right)$ can be split into two factors, by splitting its middle term and using the identity $(\mathrm{x}+\mathrm{a})(\mathrm{x}+\mathrm{b})={\mathrm{x}}^2+(\mathrm{a}+\mathrm{b})\mathrm{x}+\mathrm{ab}$.

Comparing these two, we get $(\mathrm{a}+\mathrm{b})=-7 \mathrm{and} \mathrm{ab}=10$ .

From the above condition, we can get, $\mathrm{a}=-5 \mathrm{and} \mathrm{b}=-2$

$\begin{array}{rcl}\left({\mathrm{y}}^2-7\mathrm{y}+10\right)÷(\mathrm{y}-5)& =& \frac{\left({\mathrm{y}}^2-7\mathrm{y}+10\right)}{(\mathrm{y}-5)}\\ & =& \frac{\left({\mathrm{y}}^2-5\mathrm{y}-2\mathrm{y}+10\right)}{\left(\mathrm{y}-5\right)}\\ & =& \frac{\mathrm{y}(\mathrm{y}-5)-2(\mathrm{y}-5)}{\left(\mathrm{y}-5\right)}\\ & =& \frac{(\mathrm{y}-5)(\mathrm{y}-2)}{\left(\mathrm{y}-5\right)}\\ & =& \frac{\cancel{(\mathrm{y}-5)}(\mathrm{y}-2)}{\cancel{\left(\mathrm{y}-5\right)}}\\ & =& (\mathrm{y}-2)\end{array}$