How do you factor by grouping 35xy-5x-56y+8?

May 10, 2015

Notice that $\frac{35}{5} = 7$ and $\frac{56}{8} = 7$, so we will do well to group as follows:

$35 x y - 5 x - 56 y + 8 = \left(35 x y - 5 x\right) - \left(56 y - 8\right)$
$= 5 x \left(7 y - 1\right) - 8 \left(7 y - 1\right)$
$= \left(5 x - 8\right) \left(7 y - 1\right)$.

May 10, 2015

First note that the first group of two terms have a common factor of $5 x$, so $35 x y - 5 x - 56 y + 8 = 5 x \left(7 y - 1\right) - 56 y + 8$. Next, factor $- 8$ out of the second group of two terms to get $5 x \left(7 y - 1\right) - 8 \left(7 y - 1\right)$. Now both of these terms have a factor of $\left(7 y - 1\right)$ that can be factored out. The final answer is $35 x y - 5 x - 56 y + 8 = \left(7 y - 1\right) \left(5 x - 8\right)$.

May 10, 2015

Note that the ratio between the first two coefficients ($35 : 5$) is the same as the ratio between the last two coefficients ($56 : 8$).
This gives us a hint as to a possible solution. (note that at this point it is only "possible"; it is not guaranteed).

$\textcolor{red}{\left(35 x y - 5 x\right)} - \textcolor{b l u e}{\left(56 y - 8\right)}$

$= \textcolor{red}{\left(5 x\right) \left(7 y - 1\right)} - \textcolor{b l u e}{\left(8\right) \left(7 y - 1\right)}$

$= \left(5 x - 8\right) \left(7 y - 1\right)$