Question 168695


Looking at {{{x^4-13x^2+36}}} we can see that the first term is {{{1x^4}}} and the last term is {{{36}}} where the coefficients are 1 and 36 respectively.


Now multiply the first coefficient 1 and the last coefficient 36 to get 36. Now what two numbers multiply to 36 and add to the  middle coefficient -13? Let's list all of the factors of 36:




Factors of 36:

1,2,3,4,6,9,12,18


-1,-2,-3,-4,-6,-9,-12,-18 ...List the negative factors as well. This will allow us to find all possible combinations


These factors pair up and multiply to 36

1*36

2*18

3*12

4*9

6*6

(-1)*(-36)

(-2)*(-18)

(-3)*(-12)

(-4)*(-9)

(-6)*(-6)


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



Now which of these pairs add to -13? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -13


<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td align="center">1</td><td align="center">36</td><td>1+36=37</td></tr><tr><td align="center">2</td><td align="center">18</td><td>2+18=20</td></tr><tr><td align="center">3</td><td align="center">12</td><td>3+12=15</td></tr><tr><td align="center">4</td><td align="center">9</td><td>4+9=13</td></tr><tr><td align="center">6</td><td align="center">6</td><td>6+6=12</td></tr><tr><td align="center">-1</td><td align="center">-36</td><td>-1+(-36)=-37</td></tr><tr><td align="center">-2</td><td align="center">-18</td><td>-2+(-18)=-20</td></tr><tr><td align="center">-3</td><td align="center">-12</td><td>-3+(-12)=-15</td></tr><tr><td align="center">-4</td><td align="center">-9</td><td>-4+(-9)=-13</td></tr><tr><td align="center">-6</td><td align="center">-6</td><td>-6+(-6)=-12</td></tr></table>



From this list we can see that -4 and -9 add up to -13 and multiply to 36



Now looking at the expression {{{1x^4-13x^2+36}}}, replace {{{-13x^2}}} with {{{-4x^2+-9x^2}}} (notice {{{-4x^2+-9x^2}}} adds up to {{{-13x^2}}}. So it is equivalent to {{{-13x^2}}})


{{{x^4+highlight(-4x^2+-9x^2)+36}}}



Now let's factor {{{x^4-4x^2-9x^2+36}}} by grouping:



{{{(x^4-4x^2)+(-9x^2+36)}}} Group like terms



{{{x^2(x^2-4)-9(x^2-4)}}} Factor out the GCF of {{{x^2}}} out of the first group. Factor out the GCF of {{{-9}}} out of the second group



{{{(x^2-9)(x^2-4)}}} Since we have a common term of {{{x^2-4}}}, we can combine like terms


So {{{x^4-4x^2-9x^2+36}}} factors to {{{(x^2-9)(x^2-4)}}}



So this also means that {{{x^4-13x^2+36}}} factors to {{{(x^2-9)(x^2-4)}}} (since {{{x^4-13x^2+36}}} is equivalent to {{{x^4-4x^2-9x^2+36}}})




So {{{x^4-13x^2+36}}} factors to {{{(x^2-9)(x^2-4)}}}


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{{{(x^2-9)(x^2-4)}}} Start with the given factorization



{{{(x+3)(x-3)(x^2-4)}}} Factor {{{x^2-9}}} to get {{{(x+3)(x-3)}}} (by use of the difference of squares)



{{{(x+3)(x-3)(x+2)(x-2)}}} Factor {{{x^2-4}}} to get {{{(x+2)(x-2)}}} (by use of the difference of squares)




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Answer:



So {{{x^4-13x^2+36}}} completely factors to {{{(x+3)(x-3)(x+2)(x-2)}}} 



Note: the order of the factors does not matter.