Question 143169
# 1




Looking at {{{1x^2-5x-6}}} we can see that the first term is {{{1x^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 {{{1x^2-5x-6}}}, replace {{{-5x}}} with {{{1x+-6x}}} (notice {{{1x+-6x}}} adds up to {{{-5x}}}. So it is equivalent to {{{-5x}}})


{{{1x^2+highlight(1x+-6x)+-6}}}



Now let's factor {{{1x^2+1x-6x-6}}} by grouping:



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



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




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     Answer:

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



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# 2





{{{4x^2-8x+4}}} Start with the given expression



{{{4(x^2-2x+1)}}} Factor out the GCF {{{4}}}



Now let's focus on the inner expression {{{x^2-2x+1}}}





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


Now multiply the first coefficient 1 and the last coefficient 1 to get 1. Now what two numbers multiply to 1 and add to the  middle coefficient -2? Let's list all of the factors of 1:




Factors of 1:

1


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


These factors pair up and multiply to 1

1*1

(-1)*(-1)


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



Now which of these pairs add to -2? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -2


<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td align="center">1</td><td align="center">1</td><td>1+1=2</td></tr><tr><td align="center">-1</td><td align="center">-1</td><td>-1+(-1)=-2</td></tr></table>



From this list we can see that -1 and -1 add up to -2 and multiply to 1



Now looking at the expression {{{1x^2-2x+1}}}, replace {{{-2x}}} with {{{-1x+-1x}}} (notice {{{-1x+-1x}}} adds up to {{{-2x}}}. So it is equivalent to {{{-2x}}})


{{{1x^2+highlight(-1x+-1x)+1}}}



Now let's factor {{{1x^2-1x-1x+1}}} by grouping:



{{{(1x^2-1x)+(-1x+1)}}} Group like terms



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



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


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



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



note:  {{{(x-1)(x-1)}}} is equivalent to  {{{(x-1)^2}}} since the term {{{x-1}}} occurs twice. So {{{1x^2-2x+1}}} also factors to {{{(x-1)^2}}}




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So our expression goes from {{{4(x^2-2x+1)}}} and factors further to {{{4(x-1)^2}}}



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Answer:


So {{{4x^2-8x+4}}} factors to {{{4(x-1)^2}}}