Question 282672


{{{30t^6+25t^5-30t^4}}} Start with the given expression



{{{5t^4(6t^2+5t-6)}}} Factor out the GCF {{{5t^4}}}



Now let's focus on the inner expression {{{6t^2+5t-6}}}





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Looking at {{{6t^2+5t-6}}} we can see that the first term is {{{6t^2}}} and the last term is {{{-6}}} where the coefficients are 6 and -6 respectively.


Now multiply the first coefficient 6 and the last coefficient -6 to get -36. Now what two numbers multiply to -36 and add to the  middle coefficient 5? 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)

(-1)*(36)

(-2)*(18)

(-3)*(12)

(-4)*(9)


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



Now which of these pairs add to 5? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 5


<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)=-35</td></tr><tr><td align="center">2</td><td align="center">-18</td><td>2+(-18)=-16</td></tr><tr><td align="center">3</td><td align="center">-12</td><td>3+(-12)=-9</td></tr><tr><td align="center">4</td><td align="center">-9</td><td>4+(-9)=-5</td></tr><tr><td align="center">-1</td><td align="center">36</td><td>-1+36=35</td></tr><tr><td align="center">-2</td><td align="center">18</td><td>-2+18=16</td></tr><tr><td align="center">-3</td><td align="center">12</td><td>-3+12=9</td></tr><tr><td align="center">-4</td><td align="center">9</td><td>-4+9=5</td></tr></table>



From this list we can see that -4 and 9 add up to 5 and multiply to -36



Now looking at the expression {{{6t^2+5t-6}}}, replace {{{5t}}} with {{{-4t+9t}}} (notice {{{-4t+9t}}} adds up to {{{5t}}}. So it is equivalent to {{{5t}}})


{{{6t^2+highlight(-4t+9t)+-6}}}



Now let's factor {{{6t^2-4t+9t-6}}} by grouping:



{{{(6t^2-4t)+(9t-6)}}} Group like terms



{{{2t(3t-2)+3(3t-2)}}} Factor out the GCF of {{{2t}}} out of the first group. Factor out the GCF of {{{3}}} out of the second group



{{{(2t+3)(3t-2)}}} Since we have a common term of {{{3t-2}}}, we can combine like terms


So {{{6t^2-4t+9t-6}}} factors to {{{(2t+3)(3t-2)}}}



So this also means that {{{6t^2+5t-6}}} factors to {{{(2t+3)(3t-2)}}} (since {{{6t^2+5t-6}}} is equivalent to {{{6t^2-4t+9t-6}}})




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So our expression goes from {{{5t^4(6t^2+5t-6)}}} and factors further to {{{5t^4(2t+3)(3t-2)}}}



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Answer:


So {{{30t^6+25t^5-30t^4}}} completely factors to {{{5t^4(2t+3)(3t-2)}}}