Question 145894


Looking at {{{7x^2+58x+16}}} we can see that the first term is {{{7x^2}}} and the last term is {{{16}}} where the coefficients are 7 and 16 respectively.


Now multiply the first coefficient 7 and the last coefficient 16 to get 112. Now what two numbers multiply to 112 and add to the  middle coefficient 58? Let's list all of the factors of 112:




Factors of 112:

1,2,4,7,8,14,16,28,56,112


-1,-2,-4,-7,-8,-14,-16,-28,-56,-112 ...List the negative factors as well. This will allow us to find all possible combinations


These factors pair up and multiply to 112

1*112

2*56

4*28

7*16

8*14

(-1)*(-112)

(-2)*(-56)

(-4)*(-28)

(-7)*(-16)

(-8)*(-14)


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



Now which of these pairs add to 58? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 58


<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td align="center">1</td><td align="center">112</td><td>1+112=113</td></tr><tr><td align="center">2</td><td align="center">56</td><td>2+56=58</td></tr><tr><td align="center">4</td><td align="center">28</td><td>4+28=32</td></tr><tr><td align="center">7</td><td align="center">16</td><td>7+16=23</td></tr><tr><td align="center">8</td><td align="center">14</td><td>8+14=22</td></tr><tr><td align="center">-1</td><td align="center">-112</td><td>-1+(-112)=-113</td></tr><tr><td align="center">-2</td><td align="center">-56</td><td>-2+(-56)=-58</td></tr><tr><td align="center">-4</td><td align="center">-28</td><td>-4+(-28)=-32</td></tr><tr><td align="center">-7</td><td align="center">-16</td><td>-7+(-16)=-23</td></tr><tr><td align="center">-8</td><td align="center">-14</td><td>-8+(-14)=-22</td></tr></table>



From this list we can see that 2 and 56 add up to 58 and multiply to 112



Now looking at the expression {{{7x^2+58x+16}}}, replace {{{58x}}} with {{{2x+56x}}} (notice {{{2x+56x}}} adds up to {{{58x}}}. So it is equivalent to {{{58x}}})


{{{7x^2+highlight(2x+56x)+16}}}



Now let's factor {{{7x^2+2x+56x+16}}} by grouping:



{{{(7x^2+2x)+(56x+16)}}} Group like terms



{{{x(7x+2)+8(7x+2)}}} Factor out the GCF of {{{x}}} out of the first group. Factor out the GCF of {{{8}}} out of the second group



{{{(x+8)(7x+2)}}} Since we have a common term of {{{7x+2}}}, we can combine like terms


So {{{7x^2+2x+56x+16}}} factors to {{{(x+8)(7x+2)}}}



So this also means that {{{7x^2+58x+16}}} factors to {{{(x+8)(7x+2)}}} (since {{{7x^2+58x+16}}} is equivalent to {{{7x^2+2x+56x+16}}})



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Answer:


So {{{7x^2+58x+16}}} factors to {{{(x+8)(7x+2)}}}