Question 145795
# 1




Looking at {{{x^2-5x-6}}} we can see that the first term is {{{x^2}}} and the last term is {{{-6}}} where the coefficients are 1 and -6 respectively.


Now multiply the first coefficient 1 and the last coefficient -6 to get -6. Now what two numbers multiply to -6 and add to the  middle coefficient -5? Let's list all of the factors of -6:




Factors of -6:

1,2,3,6


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


These factors pair up and multiply to -6

(1)*(-6)

(2)*(-3)

(-1)*(6)

(-2)*(3)


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">-6</td><td>1+(-6)=-5</td></tr><tr><td align="center">2</td><td align="center">-3</td><td>2+(-3)=-1</td></tr><tr><td align="center">-1</td><td align="center">6</td><td>-1+6=5</td></tr><tr><td align="center">-2</td><td align="center">3</td><td>-2+3=1</td></tr></table>



From this list we can see that 1 and -6 add up to -5 and multiply to -6



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


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



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



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



{{{x(x+1)-6(x+1)}}} Factor out the GCF of {{{x}}} out of the first group. Factor out the GCF of {{{-6}}} out of the second group



{{{(x-6)(x+1)}}} Since we have a common term of {{{x+1}}}, we can combine like terms


So {{{x^2+x-6x-6}}} factors to {{{(x-6)(x+1)}}}



So this also means that {{{x^2-5x-6}}} factors to {{{(x-6)(x+1)}}} (since {{{x^2-5x-6}}} is equivalent to {{{x^2+x-6x-6}}})




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     Answer:

So {{{x^2-5x-6}}} factors to {{{(x-6)(x+1)}}}






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# 2





Looking at {{{x^2-x-20}}} we can see that the first term is {{{x^2}}} and the last term is {{{-20}}} where the coefficients are 1 and -20 respectively.


Now multiply the first coefficient 1 and the last coefficient -20 to get -20. Now what two numbers multiply to -20 and add to the  middle coefficient -1? Let's list all of the factors of -20:




Factors of -20:

1,2,4,5,10,20


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


These factors pair up and multiply to -20

(1)*(-20)

(2)*(-10)

(4)*(-5)

(-1)*(20)

(-2)*(10)

(-4)*(5)


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



Now which of these pairs add to -1? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -1


<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td align="center">1</td><td align="center">-20</td><td>1+(-20)=-19</td></tr><tr><td align="center">2</td><td align="center">-10</td><td>2+(-10)=-8</td></tr><tr><td align="center">4</td><td align="center">-5</td><td>4+(-5)=-1</td></tr><tr><td align="center">-1</td><td align="center">20</td><td>-1+20=19</td></tr><tr><td align="center">-2</td><td align="center">10</td><td>-2+10=8</td></tr><tr><td align="center">-4</td><td align="center">5</td><td>-4+5=1</td></tr></table>



From this list we can see that 4 and -5 add up to -1 and multiply to -20



Now looking at the expression {{{x^2-x-20}}}, replace {{{-x}}} with {{{4x+-5x}}} (notice {{{4x+-5x}}} adds up to {{{-x}}}. So it is equivalent to {{{-x}}})


{{{x^2+highlight(4x+-5x)+-20}}}



Now let's factor {{{x^2+4x-5x-20}}} by grouping:



{{{(x^2+4x)+(-5x-20)}}} Group like terms



{{{x(x+4)-5(x+4)}}} Factor out the GCF of {{{x}}} out of the first group. Factor out the GCF of {{{-5}}} out of the second group



{{{(x-5)(x+4)}}} Since we have a common term of {{{x+4}}}, we can combine like terms


So {{{x^2+4x-5x-20}}} factors to {{{(x-5)(x+4)}}}



So this also means that {{{x^2-x-20}}} factors to {{{(x-5)(x+4)}}} (since {{{x^2-x-20}}} is equivalent to {{{x^2+4x-5x-20}}})




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     Answer:

So {{{x^2-x-20}}} factors to {{{(x-5)(x+4)}}}