Question 127518


Looking at {{{4x^2-33x+65}}} we can see that the first term is {{{4x^2}}} and the last term is {{{65}}} where the coefficients are 4 and 65 respectively.


Now multiply the first coefficient 4 and the last coefficient 65 to get 260. Now what two numbers multiply to 260 and add to the  middle coefficient -33? Let's list all of the factors of 260:




Factors of 260:

1,2,4,5,10,13,20,26,52,65,130,260


-1,-2,-4,-5,-10,-13,-20,-26,-52,-65,-130,-260 ...List the negative factors as well. This will allow us to find all possible combinations


These factors pair up and multiply to 260

1*260

2*130

4*65

5*52

10*26

13*20

(-1)*(-260)

(-2)*(-130)

(-4)*(-65)

(-5)*(-52)

(-10)*(-26)

(-13)*(-20)


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



Now which of these pairs add to -33? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -33


<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td align="center">1</td><td align="center">260</td><td>1+260=261</td></tr><tr><td align="center">2</td><td align="center">130</td><td>2+130=132</td></tr><tr><td align="center">4</td><td align="center">65</td><td>4+65=69</td></tr><tr><td align="center">5</td><td align="center">52</td><td>5+52=57</td></tr><tr><td align="center">10</td><td align="center">26</td><td>10+26=36</td></tr><tr><td align="center">13</td><td align="center">20</td><td>13+20=33</td></tr><tr><td align="center">-1</td><td align="center">-260</td><td>-1+(-260)=-261</td></tr><tr><td align="center">-2</td><td align="center">-130</td><td>-2+(-130)=-132</td></tr><tr><td align="center">-4</td><td align="center">-65</td><td>-4+(-65)=-69</td></tr><tr><td align="center">-5</td><td align="center">-52</td><td>-5+(-52)=-57</td></tr><tr><td align="center">-10</td><td align="center">-26</td><td>-10+(-26)=-36</td></tr><tr><td align="center">-13</td><td align="center">-20</td><td>-13+(-20)=-33</td></tr></table>



From this list we can see that -13 and -20 add up to -33 and multiply to 260



Now looking at the expression {{{4x^2-33x+65}}}, replace {{{-33x}}} with {{{-13x+-20x}}} (notice {{{-13x+-20x}}} adds up to {{{-33x}}}. So it is equivalent to {{{-33x}}})


{{{4x^2+highlight(-13x+-20x)+65}}}



Now let's factor {{{4x^2-13x-20x+65}}} by grouping:



{{{(4x^2-13x)+(-20x+65)}}} Group like terms



{{{x(4x-13)-5(4x-13)}}} Factor out the GCF of {{{x}}} out of the first group. Factor out the GCF of {{{-5}}} out of the second group



{{{(x-5)(4x-13)}}} Since we have a common term of {{{4x-13}}}, we can combine like terms


So {{{4x^2-13x-20x+65}}} factors to {{{(x-5)(4x-13)}}}



So this also means that {{{4x^2-33x+65}}} factors to {{{(x-5)(4x-13)}}} (since {{{4x^2-33x+65}}} is equivalent to {{{4x^2-13x-20x+65}}})





So {{{4x^2-33x+65}}} factors to {{{(x-5)(4x-13)}}}




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Answer:



So {{{x-5}}} is a factor of {{{4x^2-33x+65}}}