Question 150639


Looking at {{{1a^2-2ab-15b^2}}} we can see that the first term is {{{1a^2}}} and the last term is {{{-15b^2}}} where the coefficients are 1 and -15 respectively.


Now multiply the first coefficient 1 and the last coefficient -15 to get -15. Now what two numbers multiply to -15 and add to the  middle coefficient -2? Let's list all of the factors of -15:




Factors of -15:

1,3,5,15


-1,-3,-5,-15 ...List the negative factors as well. This will allow us to find all possible combinations


These factors pair up and multiply to -15

(1)*(-15)

(3)*(-5)

(-1)*(15)

(-3)*(5)


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">-15</td><td>1+(-15)=-14</td></tr><tr><td align="center">3</td><td align="center">-5</td><td>3+(-5)=-2</td></tr><tr><td align="center">-1</td><td align="center">15</td><td>-1+15=14</td></tr><tr><td align="center">-3</td><td align="center">5</td><td>-3+5=2</td></tr></table>



From this list we can see that 3 and -5 add up to -2 and multiply to -15



Now looking at the expression {{{1a^2-2ab-15b^2}}}, replace {{{-2ab}}} with {{{3ab+-5ab}}} (notice {{{3ab+-5ab}}} adds up to {{{-2ab}}}. So it is equivalent to {{{-2ab}}})


{{{1a^2+highlight(3ab+-5ab)+-15b^2}}}



Now let's factor {{{1a^2+3ab-5ab-15b^2}}} by grouping:



{{{(1a^2+3ab)+(-5ab-15b^2)}}} Group like terms



{{{a(a+3b)-5b(a+3b)}}} Factor out the GCF of {{{a}}} out of the first group. Factor out the GCF of {{{-5b}}} out of the second group



{{{(a-5b)(a+3b)}}} Since we have a common term of {{{a+3b}}}, we can combine like terms


So {{{1a^2+3ab-5ab-15b^2}}} factors to {{{(a-5b)(a+3b)}}}



So this also means that {{{1a^2-2ab-15b^2}}} factors to {{{(a-5b)(a+3b)}}} (since {{{1a^2-2ab-15b^2}}} is equivalent to {{{1a^2+3ab-5ab-15b^2}}})




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     Answer:

So {{{a^2-2ab-15b^2}}} factors to {{{(a-5b)(a+3b)}}}