Question 611194


Looking at the expression {{{5z^2-23z-10}}}, we can see that the first coefficient is {{{5}}}, the second coefficient is {{{-23}}}, and the last term is {{{-10}}}.



Now multiply the first coefficient {{{5}}} by the last term {{{-10}}} to get {{{(5)(-10)=-50}}}.



Now the question is: what two whole numbers multiply to {{{-50}}} (the previous product) <font size=4><b>and</b></font> add to the second coefficient {{{-23}}}?



To find these two numbers, we need to list <font size=4><b>all</b></font> of the factors of {{{-50}}} (the previous product).



Factors of {{{-50}}}:

1,2,5,10,25,50

-1,-2,-5,-10,-25,-50



Note: list the negative of each factor. This will allow us to find all possible combinations.



These factors pair up and multiply to {{{-50}}}.

1*(-50) = -50
2*(-25) = -50
5*(-10) = -50
(-1)*(50) = -50
(-2)*(25) = -50
(-5)*(10) = -50


Now let's add up each pair of factors to see if one pair adds to the middle coefficient {{{-23}}}:



<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td  align="center"><font color=black>1</font></td><td  align="center"><font color=black>-50</font></td><td  align="center"><font color=black>1+(-50)=-49</font></td></tr><tr><td  align="center"><font color=red>2</font></td><td  align="center"><font color=red>-25</font></td><td  align="center"><font color=red>2+(-25)=-23</font></td></tr><tr><td  align="center"><font color=black>5</font></td><td  align="center"><font color=black>-10</font></td><td  align="center"><font color=black>5+(-10)=-5</font></td></tr><tr><td  align="center"><font color=black>-1</font></td><td  align="center"><font color=black>50</font></td><td  align="center"><font color=black>-1+50=49</font></td></tr><tr><td  align="center"><font color=black>-2</font></td><td  align="center"><font color=black>25</font></td><td  align="center"><font color=black>-2+25=23</font></td></tr><tr><td  align="center"><font color=black>-5</font></td><td  align="center"><font color=black>10</font></td><td  align="center"><font color=black>-5+10=5</font></td></tr></table>



From the table, we can see that the two numbers {{{2}}} and {{{-25}}} add to {{{-23}}} (the middle coefficient).



So the two numbers {{{2}}} and {{{-25}}} both multiply to {{{-50}}} <font size=4><b>and</b></font> add to {{{-23}}}



Now replace the middle term {{{-23z}}} with {{{2z-25z}}}. Remember, {{{2}}} and {{{-25}}} add to {{{-23}}}. So this shows us that {{{2z-25z=-23z}}}.



{{{5z^2+highlight(2z-25z)-10}}} Replace the second term {{{-23z}}} with {{{2z-25z}}}.



{{{(5z^2+2z)+(-25z-10)}}} Group the terms into two pairs.



{{{z(5z+2)+(-25z-10)}}} Factor out the GCF {{{z}}} from the first group.



{{{z(5z+2)-5(5z+2)}}} Factor out {{{5}}} from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.



{{{(z-5)(5z+2)}}} Combine like terms. Or factor out the common term {{{5z+2}}}



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Answer:



So {{{5z^2-23z-10}}} factors to {{{(z-5)(5z+2)}}}.



In other words, {{{5z^2-23z-10=(z-5)(5z+2)}}}.



Note: you can check the answer by expanding {{{(z-5)(5z+2)}}} to get {{{5z^2-23z-10}}} or by graphing the original expression and the answer (the two graphs should be identical).


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