Question 194980
{{{4s^3+s^6-45}}} Start with the given expression.



{{{s^6+4s^3-45}}} Rearrange the terms.



Looking at {{{s^6+4s^3-45}}} we can see that the first term is {{{s^6}}} and the last term is {{{-45}}} where the coefficients are 1 and -45 respectively.


Now multiply the first coefficient 1 and the last coefficient -45 to get -45. Now what two numbers multiply to -45 and add to the  middle coefficient 4? Let's list all of the factors of -45:




Factors of -45:

1,3,5,9,15,45


-1,-3,-5,-9,-15,-45 ...List the negative factors as well. This will allow us to find all possible combinations


These factors pair up and multiply to -45

(1)*(-45)

(3)*(-15)

(5)*(-9)

(-1)*(45)

(-3)*(15)

(-5)*(9)


note: remember, the product of a negative and a positive number is a negative number



Now which of these pairs add to 4? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 4


<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td align="center">1</td><td align="center">-45</td><td>1+(-45)=-44</td></tr><tr><td align="center">3</td><td align="center">-15</td><td>3+(-15)=-12</td></tr><tr><td align="center">5</td><td align="center">-9</td><td>5+(-9)=-4</td></tr><tr><td align="center">-1</td><td align="center">45</td><td>-1+45=44</td></tr><tr><td align="center">-3</td><td align="center">15</td><td>-3+15=12</td></tr><tr><td align="center">-5</td><td align="center">9</td><td>-5+9=4</td></tr></table>



From this list we can see that -5 and 9 add up to 4 and multiply to -45



Now looking at the expression {{{s^6+4s^3-45}}}, replace {{{4s^3}}} with {{{-5s^3+9s^3}}} (notice {{{-5s^3+9s^3}}} adds up to {{{4s^3}}}. So it is equivalent to {{{4s^3}}})


{{{s^6+highlight(-5s^3+9s^3)-45}}}



Now let's factor {{{s^6-5s^3+9s^3-45}}} by grouping:



{{{(s^6-5s^3)+(9s^3-45)}}} Group like terms



{{{s^3(s^3-5)+9(s^3-5)}}} Factor out the GCF of {{{s^3}}} out of the first group. Factor out the GCF of {{{9}}} out of the second group



{{{(s^3+9)(s^3-5)}}} Since we have a common term of {{{s^3-5}}}, we can combine like terms


So {{{s^6-5s^3+9s^3-45}}} factors to {{{(s^3+9)(s^3-5)}}}



So this also means that {{{s^6+4s^3-45}}} factors to {{{(s^3+9)(s^3-5)}}} (since {{{s^6+4s^3-45}}} is equivalent to {{{s^6-5s^3+9s^3-45}}})




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     Answer:

So {{{4s^3+s^6-45}}} completely factors to {{{(s^3+9)(s^3-5)}}}