Question 126662


Looking at {{{3t^2+14t-5}}} we can see that the first term is {{{3t^2}}} and the last term is {{{-5}}} where the coefficients are 3 and -5 respectively.


Now multiply the first coefficient 3 and the last coefficient -5 to get -15. Now what two numbers multiply to -15 and add to the  middle coefficient 14? Let's list all of the factors of -15:




Factors of -15:

1,3,5,15


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


These factors pair up and multiply to -15

(1)*(-15)

(3)*(-5)

(-1)*(15)

(-3)*(5)


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



Now which of these pairs add to 14? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 14


<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td align="center">1</td><td align="center">-15</td><td>1+(-15)=-14</td></tr><tr><td align="center">3</td><td align="center">-5</td><td>3+(-5)=-2</td></tr><tr><td align="center">-1</td><td align="center">15</td><td>-1+15=14</td></tr><tr><td align="center">-3</td><td align="center">5</td><td>-3+5=2</td></tr></table>



From this list we can see that -1 and 15 add up to 14 and multiply to -15



Now looking at the expression {{{3t^2+14t-5}}}, replace {{{14t}}} with {{{-1t+15t}}} (notice {{{-1t+15t}}} adds up to {{{14t}}}. So it is equivalent to {{{14t}}})


{{{3t^2+highlight(-1t+15t)+-5}}}



Now let's factor {{{3t^2-1t+15t-5}}} by grouping:



{{{(3t^2-1t)+(15t-5)}}} Group like terms



{{{t(3t-1)+5(3t-1)}}} Factor out the GCF of {{{t}}} out of the first group. Factor out the GCF of {{{5}}} out of the second group



{{{(t+5)(3t-1)}}} Since we have a common term of {{{3t-1}}}, we can combine like terms


So {{{3t^2-1t+15t-5}}} factors to {{{(t+5)(3t-1)}}}



So this also means that {{{3t^2+14t-5}}} factors to {{{(t+5)(3t-1)}}} (since {{{3t^2+14t-5}}} is equivalent to {{{3t^2-1t+15t-5}}})




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     Answer:

So {{{3t^2+14t-5}}} factors to {{{(t+5)(3t-1)}}}