Question 142171


Looking at {{{3z^2-10z-25}}} we can see that the first term is {{{3z^2}}} and the last term is {{{-25}}} where the coefficients are 3 and -25 respectively.


Now multiply the first coefficient 3 and the last coefficient -25 to get -75. Now what two numbers multiply to -75 and add to the  middle coefficient -10? Let's list all of the factors of -75:




Factors of -75:

1,3,5,15,25,75


-1,-3,-5,-15,-25,-75 ...List the negative factors as well. This will allow us to find all possible combinations


These factors pair up and multiply to -75

(1)*(-75)

(3)*(-25)

(5)*(-15)

(-1)*(75)

(-3)*(25)

(-5)*(15)


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



Now which of these pairs add to -10? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -10


<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td align="center">1</td><td align="center">-75</td><td>1+(-75)=-74</td></tr><tr><td align="center">3</td><td align="center">-25</td><td>3+(-25)=-22</td></tr><tr><td align="center">5</td><td align="center">-15</td><td>5+(-15)=-10</td></tr><tr><td align="center">-1</td><td align="center">75</td><td>-1+75=74</td></tr><tr><td align="center">-3</td><td align="center">25</td><td>-3+25=22</td></tr><tr><td align="center">-5</td><td align="center">15</td><td>-5+15=10</td></tr></table>



From this list we can see that 5 and -15 add up to -10 and multiply to -75



Now looking at the expression {{{3z^2-10z-25}}}, replace {{{-10z}}} with {{{5z+-15z}}} (notice {{{5z+-15z}}} adds up to {{{-10z}}}. So it is equivalent to {{{-10z}}})


{{{3z^2+highlight(5z+-15z)+-25}}}



Now let's factor {{{3z^2+5z-15z-25}}} by grouping:



{{{(3z^2+5z)+(-15z-25)}}} Group like terms



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



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


So {{{3z^2+5z-15z-25}}} factors to {{{(z-5)(3z+5)}}}



So this also means that {{{3z^2-10z-25}}} factors to {{{(z-5)(3z+5)}}} (since {{{3z^2-10z-25}}} is equivalent to {{{3z^2+5z-15z-25}}})



-------------------------------

Answer:


So {{{3z^2-10z-25}}} factors to {{{(z-5)(3z+5)}}}