Question 251559


{{{64a^4-224a^2+196}}} Start with the given expression



{{{4(16a^4-56a^2+49)}}} Factor out the GCF {{{4}}}



Now let's focus on the inner expression {{{16a^4-56a^2+49}}}





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Looking at {{{16a^4-56a^2+49}}} we can see that the first term is {{{16a^4}}} and the last term is {{{49}}} 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 {{{16a^4-56a^2+49}}}, replace {{{-56a^2}}} with {{{-28a^2-28a^2}}} (notice {{{-28a^2-28a^2}}} combines back to {{{-56a^2}}}. So it is equivalent to {{{-56a^2}}})


{{{16a^4+highlight(-28a^2-28a^2)+49}}}



Now let's factor {{{16a^4-28a^2-28a^2+49}}} by grouping:



{{{(16a^4-28a^2)+(-28a^2+49)}}} Group like terms



{{{4a^2(4a^2-7)-7(4a^2-7)}}} Factor out the GCF of {{{4a^2}}} out of the first group. Factor out the GCF of {{{-7}}} out of the second group



{{{(4a^2-7)(4a^2-7)}}} Since we have a common term of {{{4a^2-7}}}, we can combine like terms


So {{{16a^4-28a^2-28a^2+49}}} factors to {{{(4a^2-7)(4a^2-7)}}}



So this also means that {{{16a^4-56a^2+49}}} factors to {{{(4a^2-7)(4a^2-7)}}} (since {{{16a^4-56a^2+49}}} is equivalent to {{{16a^4-28a^2-28a^2+49}}})



note:  {{{(4a^2-7)(4a^2-7)}}} is equivalent to  {{{(4a^2-7)^2}}} since the term {{{4a^2-7}}} occurs twice. So {{{16a^4-56a^2+49}}} also factors to {{{(4a^2-7)^2}}}




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So our expression goes from {{{4(16a^4-56a^2+49)}}} and factors further to {{{4(4a^2-7)^2}}}



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Answer:


So {{{64a^4-224a^2+196}}} factors to {{{4(4a^2-7)^2}}}