Question 117163


Looking at {{{x^2-11x+18}}} we can see that the first term is {{{x^2}}} and the last term is {{{18}}} where the coefficients are 1 and 18 respectively.


Now multiply the first coefficient 1 and the last coefficient 18 to get 18. Now what two numbers multiply to 18 and add to the  middle coefficient -11? Let's list all of the factors of 18:




Factors of 18:

1,2,3,6,9,18


-1,-2,-3,-6,-9,-18 ...List the negative factors as well. This will allow us to find all possible combinations


These factors pair up and multiply to 18

1*18

2*9

3*6

(-1)*(-18)

(-2)*(-9)

(-3)*(-6)


note: remember two negative numbers multiplied together make a positive number



Now which of these pairs add to -11? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -11


<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td align="center">1</td><td align="center">18</td><td>1+18=19</td></tr><tr><td align="center">2</td><td align="center">9</td><td>2+9=11</td></tr><tr><td align="center">3</td><td align="center">6</td><td>3+6=9</td></tr><tr><td align="center">-1</td><td align="center">-18</td><td>-1+(-18)=-19</td></tr><tr><td align="center">-2</td><td align="center">-9</td><td>-2+(-9)=-11</td></tr><tr><td align="center">-3</td><td align="center">-6</td><td>-3+(-6)=-9</td></tr></table>



From this list we can see that -2 and -9 add up to -11 and multiply to 18



Now looking at the expression {{{x^2-11x+18}}}, replace {{{-11x}}} with {{{-2x+-9x}}} (notice {{{-2x+-9x}}} adds up to {{{-11x}}}. So it is equivalent to {{{-11x}}})


{{{x^2+highlight(-2x+-9x)+18}}}



Now let's factor {{{x^2-2x-9x+18}}} by grouping:



{{{(x^2-2x)+(-9x+18)}}} Group like terms



{{{x(x-2)-9(x-2)}}} Factor out the GCF of {{{x}}} out of the first group. Factor out the GCF of {{{-9}}} out of the second group



{{{(x-9)(x-2)}}} Since we have a common term of {{{x-2}}}, we can combine like terms


So {{{x^2-2x-9x+18}}} factors to {{{(x-9)(x-2)}}}



So this also means that {{{x^2-11x+18}}} factors to {{{(x-9)(x-2)}}} (since {{{x^2-11x+18}}} is equivalent to {{{x^2-2x-9x+18}}})


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Answer:


So {{{x^2-11x+18}}} factors to {{{(x-9)(x-2)}}}