Question 117687
#1




Looking at {{{x^2+5x-66}}} we can see that the first term is {{{x^2}}} and the last term is {{{-66}}} where the coefficients are 1 and -66 respectively.


Now multiply the first coefficient 1 and the last coefficient -66 to get -66. Now what two numbers multiply to -66 and add to the  middle coefficient 5? Let's list all of the factors of -66:




Factors of -66:

1,2,3,6,11,22,33,66


-1,-2,-3,-6,-11,-22,-33,-66 ...List the negative factors as well. This will allow us to find all possible combinations


These factors pair up and multiply to -66

(1)*(-66)

(2)*(-33)

(3)*(-22)

(6)*(-11)

(-1)*(66)

(-2)*(33)

(-3)*(22)

(-6)*(11)


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">-66</td><td>1+(-66)=-65</td></tr><tr><td align="center">2</td><td align="center">-33</td><td>2+(-33)=-31</td></tr><tr><td align="center">3</td><td align="center">-22</td><td>3+(-22)=-19</td></tr><tr><td align="center">6</td><td align="center">-11</td><td>6+(-11)=-5</td></tr><tr><td align="center">-1</td><td align="center">66</td><td>-1+66=65</td></tr><tr><td align="center">-2</td><td align="center">33</td><td>-2+33=31</td></tr><tr><td align="center">-3</td><td align="center">22</td><td>-3+22=19</td></tr><tr><td align="center">-6</td><td align="center">11</td><td>-6+11=5</td></tr></table>



From this list we can see that -6 and 11 add up to 5 and multiply to -66



Now looking at the expression {{{x^2+5x-66}}}, replace {{{5x}}} with {{{-6x+11x}}} (notice {{{-6x+11x}}} adds up to {{{5x}}}. So it is equivalent to {{{5x}}})


{{{x^2+highlight(-6x+11x)+-66}}}



Now let's factor {{{x^2-6x+11x-66}}} by grouping:



{{{(x^2-6x)+(11x-66)}}} Group like terms



{{{x(x-6)+11(x-6)}}} Factor out the GCF of {{{x}}} out of the first group. Factor out the GCF of {{{11}}} out of the second group



{{{(x+11)(x-6)}}} Since we have a common term of {{{x-6}}}, we can combine like terms


So {{{x^2-6x+11x-66}}} factors to {{{(x+11)(x-6)}}}



So this also means that {{{x^2+5x-66}}} factors to {{{(x+11)(x-6)}}} (since {{{x^2+5x-66}}} is equivalent to {{{x^2-6x+11x-66}}})


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Answer:


So {{{x^2+5x-66}}} factors to {{{(x+11)(x-6)}}}



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#2





Looking at {{{x^2-13xy+40y^2}}} we can see that the first term is {{{x^2}}} and the last term is {{{40y^2}}} where the coefficients are 1 and 40 respectively.


Now multiply the first coefficient 1 and the last coefficient 40 to get 40. Now what two numbers multiply to 40 and add to the  middle coefficient -13? Let's list all of the factors of 40:




Factors of 40:

1,2,4,5,8,10,20,40


-1,-2,-4,-5,-8,-10,-20,-40 ...List the negative factors as well. This will allow us to find all possible combinations


These factors pair up and multiply to 40

1*40

2*20

4*10

5*8

(-1)*(-40)

(-2)*(-20)

(-4)*(-10)

(-5)*(-8)


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



Now which of these pairs add to -13? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -13


<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td align="center">1</td><td align="center">40</td><td>1+40=41</td></tr><tr><td align="center">2</td><td align="center">20</td><td>2+20=22</td></tr><tr><td align="center">4</td><td align="center">10</td><td>4+10=14</td></tr><tr><td align="center">5</td><td align="center">8</td><td>5+8=13</td></tr><tr><td align="center">-1</td><td align="center">-40</td><td>-1+(-40)=-41</td></tr><tr><td align="center">-2</td><td align="center">-20</td><td>-2+(-20)=-22</td></tr><tr><td align="center">-4</td><td align="center">-10</td><td>-4+(-10)=-14</td></tr><tr><td align="center">-5</td><td align="center">-8</td><td>-5+(-8)=-13</td></tr></table>



From this list we can see that -5 and -8 add up to -13 and multiply to 40



Now looking at the expression {{{x^2-13xy+40y^2}}}, replace {{{-13xy}}} with {{{-5xy+-8xy}}} (notice {{{-5xy+-8xy}}} adds up to {{{-13xy}}}. So it is equivalent to {{{-13xy}}})


{{{x^2+highlight(-5xy+-8xy)+40y^2}}}



Now let's factor {{{x^2-5xy-8xy+40y^2}}} by grouping:



{{{(x^2-5xy)+(-8xy+40y^2)}}} Group like terms



{{{x(x-5y)-8y(x-5y)}}} Factor out the GCF of {{{x}}} out of the first group. Factor out the GCF of {{{-8y}}} out of the second group



{{{(x-8y)(x-5y)}}} Since we have a common term of {{{x-5y}}}, we can combine like terms


So {{{x^2-5xy-8xy+40y^2}}} factors to {{{(x-8y)(x-5y)}}}



So this also means that {{{x^2-13xy+40y^2}}} factors to {{{(x-8y)(x-5y)}}} (since {{{x^2-13xy+40y^2}}} is equivalent to {{{x^2-5xy-8xy+40y^2}}})


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Answer:


So {{{x^2-13xy+40y^2}}} factors to {{{(x-8y)(x-5y)}}}