Question 417524


Looking at {{{4x^2-17x+4}}} we can see that the first term is {{{4x^2}}} and the last term is {{{4}}} where the coefficients are 4 and 4 respectively.


Now multiply the first coefficient 4 and the last coefficient 4 to get 16. Now what two numbers multiply to 16 and add to the  middle coefficient -17? Let's list all of the factors of 16:




Factors of 16:

1,2,4,8


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


These factors pair up and multiply to 16

1*16

2*8

4*4

(-1)*(-16)

(-2)*(-8)

(-4)*(-4)


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



Now which of these pairs add to -17? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -17


<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td align="center">1</td><td align="center">16</td><td>1+16=17</td></tr><tr><td align="center">2</td><td align="center">8</td><td>2+8=10</td></tr><tr><td align="center">4</td><td align="center">4</td><td>4+4=8</td></tr><tr><td align="center">-1</td><td align="center">-16</td><td>-1+(-16)=-17</td></tr><tr><td align="center">-2</td><td align="center">-8</td><td>-2+(-8)=-10</td></tr><tr><td align="center">-4</td><td align="center">-4</td><td>-4+(-4)=-8</td></tr></table>



From this list we can see that -1 and -16 add up to -17 and multiply to 16



Now looking at the expression {{{4x^2-17x+4}}}, replace {{{-17x}}} with {{{-1x+-16x}}} (notice {{{-1x+-16x}}} adds up to {{{-17x}}}. So it is equivalent to {{{-17x}}})


{{{4x^2+highlight(-1x+-16x)+4}}}



Now let's factor {{{4x^2-1x-16x+4}}} by grouping:



{{{(4x^2-1x)+(-16x+4)}}} Group like terms



{{{x(4x-1)-4(4x-1)}}} Factor out the GCF of {{{x}}} out of the first group. Factor out the GCF of {{{-4}}} out of the second group



{{{(x-4)(4x-1)}}} Since we have a common term of {{{4x-1}}}, we can combine like terms


So {{{4x^2-1x-16x+4}}} factors to {{{(x-4)(4x-1)}}}



So this also means that {{{4x^2-17x+4}}} factors to {{{(x-4)(4x-1)}}} (since {{{4x^2-17x+4}}} is equivalent to {{{4x^2-1x-16x+4}}})




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     Answer:

So {{{4x^2-17x+4}}} factors to {{{(x-4)(4x-1)}}}



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