Question 119700


Looking at {{{z^2+18z+45}}} we can see that the first term is {{{z^2}}} 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 18? 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 two negative numbers multiplied together make a positive number



Now which of these pairs add to 18? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 18


<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=46</td></tr><tr><td align="center">3</td><td align="center">15</td><td>3+15=18</td></tr><tr><td align="center">5</td><td align="center">9</td><td>5+9=14</td></tr><tr><td align="center">-1</td><td align="center">-45</td><td>-1+(-45)=-46</td></tr><tr><td align="center">-3</td><td align="center">-15</td><td>-3+(-15)=-18</td></tr><tr><td align="center">-5</td><td align="center">-9</td><td>-5+(-9)=-14</td></tr></table>



From this list we can see that 3 and 15 add up to 18 and multiply to 45



Now looking at the expression {{{z^2+18z+45}}}, replace {{{18z}}} with {{{3z+15z}}} (notice {{{3z+15z}}} adds up to {{{18z}}}. So it is equivalent to {{{18z}}})


{{{z^2+highlight(3z+15z)+45}}}



Now let's factor {{{z^2+3z+15z+45}}} by grouping:



{{{(z^2+3z)+(15z+45)}}} Group like terms



{{{z(z+3)+15(z+3)}}} Factor out the GCF of {{{z}}} out of the first group. Factor out the GCF of {{{15}}} out of the second group



{{{(z+15)(z+3)}}} Since we have a common term of {{{z+3}}}, we can combine like terms


So {{{z^2+3z+15z+45}}} factors to {{{(z+15)(z+3)}}}



So this also means that {{{z^2+18z+45}}} factors to {{{(z+15)(z+3)}}} (since {{{z^2+18z+45}}} is equivalent to {{{z^2+3z+15z+45}}})




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     Answer:

So {{{z^2+18z+45}}} factors to {{{(z+15)(z+3)}}}