Question 173291


Looking at {{{3x^2+26xy+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 26? 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 two negative numbers multiplied together make a positive number



Now which of these pairs add to 26? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 26


<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=49</td></tr><tr><td align="center"><font color="red">2</font></td><td align="center"><font color="red">24</font></td><td><font color="red">2+24=26</font></td></tr><tr><td align="center">3</td><td align="center">16</td><td>3+16=19</td></tr><tr><td align="center">4</td><td align="center">12</td><td>4+12=16</td></tr><tr><td align="center">6</td><td align="center">8</td><td>6+8=14</td></tr><tr><td align="center">-1</td><td align="center">-48</td><td>-1+(-48)=-49</td></tr><tr><td align="center">-2</td><td align="center">-24</td><td>-2+(-24)=-26</td></tr><tr><td align="center">-3</td><td align="center">-16</td><td>-3+(-16)=-19</td></tr><tr><td align="center">-4</td><td align="center">-12</td><td>-4+(-12)=-16</td></tr><tr><td align="center">-6</td><td align="center">-8</td><td>-6+(-8)=-14</td></tr></table>



From this list we can see that 2 and 24 add up to 26 and multiply to 48



Now looking at the expression {{{3x^2+26xy+16y^2}}}, replace {{{26xy}}} with {{{2xy+24xy}}} (notice {{{2xy+24xy}}} adds up to {{{26xy}}}. So it is equivalent to {{{26xy}}})


{{{3x^2+highlight(2xy+24xy)+16y^2}}}



Now let's factor {{{3x^2+2xy+24xy+16y^2}}} by grouping:



{{{(3x^2+2xy)+(24xy+16y^2)}}} Group like terms



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



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


So {{{3x^2+2xy+24xy+16y^2}}} factors to {{{(x+8y)(3x+2y)}}}



So this also means that {{{3x^2+26xy+16y^2}}} factors to {{{(x+8y)(3x+2y)}}} (since {{{3x^2+26xy+16y^2}}} is equivalent to {{{3x^2+2xy+24xy+16y^2}}})




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     Answer:

So {{{3x^2+26xy+16y^2}}} factors to {{{(x+8y)(3x+2y)}}}