Question 316992


Looking at {{{3x^2+2xy-16y^2}}} we can see that the first term is {{{3x^2}}} and the last term is {{{-16y^2}}} where the coefficients are 3 and -16 respectively.


Now multiply the first coefficient 3 and the last coefficient -16 to get -48. Now what two numbers multiply to -48 and add to the  middle coefficient 2? Let's list all of the factors of -48:




Factors of -48:

1,2,3,4,6,8,12,16,24,48


-1,-2,-3,-4,-6,-8,-12,-16,-24,-48 ...List the negative factors as well. This will allow us to find all possible combinations


These factors pair up and multiply to -48

(1)*(-48)

(2)*(-24)

(3)*(-16)

(4)*(-12)

(6)*(-8)

(-1)*(48)

(-2)*(24)

(-3)*(16)

(-4)*(12)

(-6)*(8)


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">-48</td><td>1+(-48)=-47</td></tr><tr><td align="center">2</td><td align="center">-24</td><td>2+(-24)=-22</td></tr><tr><td align="center">3</td><td align="center">-16</td><td>3+(-16)=-13</td></tr><tr><td align="center">4</td><td align="center">-12</td><td>4+(-12)=-8</td></tr><tr><td align="center">6</td><td align="center">-8</td><td>6+(-8)=-2</td></tr><tr><td align="center">-1</td><td align="center">48</td><td>-1+48=47</td></tr><tr><td align="center">-2</td><td align="center">24</td><td>-2+24=22</td></tr><tr><td align="center">-3</td><td align="center">16</td><td>-3+16=13</td></tr><tr><td align="center">-4</td><td align="center">12</td><td>-4+12=8</td></tr><tr><td align="center">-6</td><td align="center">8</td><td>-6+8=2</td></tr></table>



From this list we can see that -6 and 8 add up to 2 and multiply to -48



Now looking at the expression {{{3x^2+2xy-16y^2}}}, replace {{{2xy}}} with {{{-6xy+8xy}}} (notice {{{-6xy+8xy}}} adds up to {{{2xy}}}. So it is equivalent to {{{2xy}}})


{{{3x^2+highlight(-6xy+8xy)+-16y^2}}}



Now let's factor {{{3x^2-6xy+8xy-16y^2}}} by grouping:



{{{(3x^2-6xy)+(8xy-16y^2)}}} Group like terms



{{{3x(x-2y)+8y(x-2y)}}} Factor out the GCF of {{{3x}}} out of the first group. Factor out the GCF of {{{8y}}} out of the second group



{{{(3x+8y)(x-2y)}}} Since we have a common term of {{{x-2y}}}, we can combine like terms


So {{{3x^2-6xy+8xy-16y^2}}} factors to {{{(3x+8y)(x-2y)}}}



So this also means that {{{3x^2+2xy-16y^2}}} factors to {{{(3x+8y)(x-2y)}}} (since {{{3x^2+2xy-16y^2}}} is equivalent to {{{3x^2-6xy+8xy-16y^2}}})




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     Answer:

So {{{3x^2+2xy-16y^2}}} factors to {{{(3x+8y)(x-2y)}}}



In other words, {{{3x^2+2xy-16y^2=(3x+8y)(x-2y)}}}