Question 191469
I'll do the first two to get you started, repost if you need more help.



# 1


{{{32k^3n^2m+112k^2nm^2+98km^3}}} Start with the given expression



{{{2km(16k^2n^2+56knm+49m^2)}}} Factor out the GCF {{{2km}}}



Now let's focus on the inner expression {{{16k^2n^2+56knm+49m^2}}}





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Looking at {{{16k^2n^2+56knm+49m^2}}} we can see that the first term is {{{16k^2n^2}}} and the last term is {{{49m^2}}} where the coefficients are 16 and 49 respectively.


Now multiply the first coefficient 16 and the last coefficient 49 to get 784. Now what two numbers multiply to 784 and add to the  middle coefficient 56? Let's list all of the factors of 784:




Factors of 784:

1,2,4,7,8,14,16,28,49,56,98,112,196,392


-1,-2,-4,-7,-8,-14,-16,-28,-49,-56,-98,-112,-196,-392 ...List the negative factors as well. This will allow us to find all possible combinations


These factors pair up and multiply to 784

1*784

2*392

4*196

7*112

8*98

14*56

16*49

28*28

(-1)*(-784)

(-2)*(-392)

(-4)*(-196)

(-7)*(-112)

(-8)*(-98)

(-14)*(-56)

(-16)*(-49)

(-28)*(-28)


note: remember two negative numbers multiplied together make a positive number



Now which of these pairs add to 56? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 56


<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td align="center">1</td><td align="center">784</td><td>1+784=785</td></tr><tr><td align="center">2</td><td align="center">392</td><td>2+392=394</td></tr><tr><td align="center">4</td><td align="center">196</td><td>4+196=200</td></tr><tr><td align="center">7</td><td align="center">112</td><td>7+112=119</td></tr><tr><td align="center">8</td><td align="center">98</td><td>8+98=106</td></tr><tr><td align="center">14</td><td align="center">56</td><td>14+56=70</td></tr><tr><td align="center">16</td><td align="center">49</td><td>16+49=65</td></tr><tr><td align="center">28</td><td align="center">28</td><td>28+28=56</td></tr><tr><td align="center">-1</td><td align="center">-784</td><td>-1+(-784)=-785</td></tr><tr><td align="center">-2</td><td align="center">-392</td><td>-2+(-392)=-394</td></tr><tr><td align="center">-4</td><td align="center">-196</td><td>-4+(-196)=-200</td></tr><tr><td align="center">-7</td><td align="center">-112</td><td>-7+(-112)=-119</td></tr><tr><td align="center">-8</td><td align="center">-98</td><td>-8+(-98)=-106</td></tr><tr><td align="center">-14</td><td align="center">-56</td><td>-14+(-56)=-70</td></tr><tr><td align="center">-16</td><td align="center">-49</td><td>-16+(-49)=-65</td></tr><tr><td align="center">-28</td><td align="center">-28</td><td>-28+(-28)=-56</td></tr></table>



From this list we can see that 28 and 28 add up to 56 and multiply to 784



Now looking at the expression {{{16k^2n^2+56knm+49m^2}}}, replace {{{56knm}}} with {{{28knm+28knm}}} (notice {{{28knm+28knm}}} adds up to {{{56knm}}}. So it is equivalent to {{{56knm}}})


{{{16k^2n^2+highlight(28knm+28knm)+49m^2}}}



Now let's factor {{{16k^2n^2+28knm+28knm+49m^2}}} by grouping:



{{{(16k^2n^2+28knm)+(28knm+49m^2)}}} Group like terms



{{{4kn(4kn+7m)+7m(4kn+7m)}}} Factor out the GCF of {{{4kn}}} out of the first group. Factor out the GCF of {{{7m}}} out of the second group



{{{(4kn+7m)(4kn+7m)}}} Combine like terms.



{{{(4kn+7m)^2}}} Condense the terms.



So {{{16k^2n^2+28knm+28knm+49m^2}}} factors to {{{(4kn+7m)^2}}}


So this also means that {{{16k^2n^2+56knm+49m^2}}} factors to {{{(4kn+7m)^2}}} (since {{{16k^2n^2+56knm+49m^2}}} is equivalent to {{{16k^2n^2+28knm+28knm+49m^2}}})




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So our expression goes from {{{2km(16k^2n^2+56knm+49m^2)}}} and factors further to {{{2km(4kn+7m)^2}}}



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Answer:


So {{{32k^3n^2m+112k^2nm^2+98km^3}}} completely factors to {{{2km(4kn+7m)^2}}}

    


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# 2



{{{9k^2-16m^2}}} Start with the given expression.



{{{(3k)^2-16m^2}}} Rewrite {{{9k^2}}} as {{{(3k)^2}}}.



{{{(3k)^2-(4m)^2}}} Rewrite {{{16m^2}}} as {{{(4m)^2}}}.



Notice how we have a difference of squares. So let's use the difference of squares formula {{{A^2-B^2=(A+B)(A-B)}}} to factor the expression:



{{{(3k+4m)(3k-4m)}}} Factor the expression using the difference of squares.



So {{{9k^2-16m^2}}} factors to {{{(3k+4m)(3k-4m)}}}.



In other words {{{9k^2-16m^2=(3k+4m)(3k-4m)}}}.