Question 146889
I'm assuming that you want to factor.





Looking at {{{35x^2-2x-1}}} we can see that the first term is {{{35x^2}}} and the last term is {{{-1}}} where the coefficients are 35 and -1 respectively.


Now multiply the first coefficient 35 and the last coefficient -1 to get -35. Now what two numbers multiply to -35 and add to the  middle coefficient -2? Let's list all of the factors of -35:




Factors of -35:

1,5,7,35


-1,-5,-7,-35 ...List the negative factors as well. This will allow us to find all possible combinations


These factors pair up and multiply to -35

(1)*(-35)

(5)*(-7)

(-1)*(35)

(-5)*(7)


note: remember, the product of a negative and a positive number is a negative 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">-35</td><td>1+(-35)=-34</td></tr><tr><td align="center">5</td><td align="center">-7</td><td>5+(-7)=-2</td></tr><tr><td align="center">-1</td><td align="center">35</td><td>-1+35=34</td></tr><tr><td align="center">-5</td><td align="center">7</td><td>-5+7=2</td></tr></table>



From this list we can see that 5 and -7 add up to -2 and multiply to -35



Now looking at the expression {{{35x^2-2x-1}}}, replace {{{-2x}}} with {{{5x+-7x}}} (notice {{{5x+-7x}}} adds up to {{{-2x}}}. So it is equivalent to {{{-2x}}})


{{{35x^2+highlight(5x+-7x)+-1}}}



Now let's factor {{{35x^2+5x-7x-1}}} by grouping:



{{{(35x^2+5x)+(-7x-1)}}} Group like terms



{{{5x(7x+1)-1(7x+1)}}} Factor out the GCF of {{{5x}}} out of the first group. Factor out the GCF of {{{-1}}} out of the second group



{{{(5x-1)(7x+1)}}} Since we have a common term of {{{7x+1}}}, we can combine like terms


So {{{35x^2+5x-7x-1}}} factors to {{{(5x-1)(7x+1)}}}



So this also means that {{{35x^2-2x-1}}} factors to {{{(5x-1)(7x+1)}}} (since {{{35x^2-2x-1}}} is equivalent to {{{35x^2+5x-7x-1}}})




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     Answer:

So {{{35x^2-2x-1}}} factors to {{{(5x-1)(7x+1)}}}