Question 112030

Looking at {{{5a^2-2ab-7b^2}}} we can see that the first term is {{{5a^2}}} and the last term is {{{-7b^2}}} where the coefficients are 5 and -7 respectively.


Now multiply the first coefficient 5 and the last coefficient -7 to get -35. Now what two numbers multiply to -35 and add to the  middle coefficient -2? Let's list all of the factors of -35:




Factors of -35:

1,5,7,35


-1,-5,-7,-35 ...List the negative factors as well. This will allow us to find all possible combinations


These factors pair up and multiply to -35

(1)*(-35)

(5)*(-7)

(-1)*(35)

(-5)*(7)


note: remember, the product of a negative and a positive number is a negative number



Now which of these pairs add to -2? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -2


<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td align="center">1</td><td align="center">-35</td><td>1+(-35)=-34</td></tr><tr><td align="center">5</td><td align="center">-7</td><td>5+(-7)=-2</td></tr><tr><td align="center">-1</td><td align="center">35</td><td>-1+35=34</td></tr><tr><td align="center">-5</td><td align="center">7</td><td>-5+7=2</td></tr></table>



From this list we can see that 5 and -7 add up to -2 and multiply to -35



Now looking at the expression {{{5a^2-2ab-7b^2}}}, replace {{{-2ab}}} with {{{5ab+-7ab}}} (notice {{{5ab+-7ab}}} adds up to {{{-2ab}}}. So it is equivalent to {{{-2ab}}})


{{{5a^2+highlight(5ab+-7ab)+-7b^2}}}



Now let's factor {{{5a^2+5ab-7ab-7b^2}}} by grouping:



{{{(5a^2+5ab)+(-7ab-7b^2)}}} Group like terms



{{{5a(a+b)-7b(a+b)}}} Factor out the GCF of {{{5a}}} out of the first group. Factor out the GCF of {{{-7b}}} out of the second group



{{{(5a-7b)(a+b)}}} Since we have a common term of {{{a+b}}}, we can combine like terms


So {{{5a^2+5ab-7ab-7b^2}}} factors to {{{(5a-7b)(a+b)}}}



So this also means that {{{5a^2-2ab-7b^2}}} factors to {{{(5a-7b)(a+b)}}} (since {{{5a^2-2ab-7b^2}}} is equivalent to {{{5a^2+5ab-7ab-7b^2}}})