Question 610467


Looking at the expression {{{p^2+11p+10}}}, we can see that the first coefficient is {{{1}}}, the second coefficient is {{{11}}}, and the last term is {{{10}}}.



Now multiply the first coefficient {{{1}}} by the last term {{{10}}} to get {{{(1)(10)=10}}}.



Now the question is: what two whole numbers multiply to {{{10}}} (the previous product) <font size=4><b>and</b></font> add to the second coefficient {{{11}}}?



To find these two numbers, we need to list <font size=4><b>all</b></font> of the factors of {{{10}}} (the previous product).



Factors of {{{10}}}:

1,2,5,10

-1,-2,-5,-10



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



These factors pair up and multiply to {{{10}}}.

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


Now let's add up each pair of factors to see if one pair adds to the middle coefficient {{{11}}}:



<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td  align="center"><font color=red>1</font></td><td  align="center"><font color=red>10</font></td><td  align="center"><font color=red>1+10=11</font></td></tr><tr><td  align="center"><font color=black>2</font></td><td  align="center"><font color=black>5</font></td><td  align="center"><font color=black>2+5=7</font></td></tr><tr><td  align="center"><font color=black>-1</font></td><td  align="center"><font color=black>-10</font></td><td  align="center"><font color=black>-1+(-10)=-11</font></td></tr><tr><td  align="center"><font color=black>-2</font></td><td  align="center"><font color=black>-5</font></td><td  align="center"><font color=black>-2+(-5)=-7</font></td></tr></table>



From the table, we can see that the two numbers {{{1}}} and {{{10}}} add to {{{11}}} (the middle coefficient).



So the two numbers {{{1}}} and {{{10}}} both multiply to {{{10}}} <font size=4><b>and</b></font> add to {{{11}}}



Now replace the middle term {{{11p}}} with {{{p+10p}}}. Remember, {{{1}}} and {{{10}}} add to {{{11}}}. So this shows us that {{{p+10p=11p}}}.



{{{p^2+highlight(p+10p)+10}}} Replace the second term {{{11p}}} with {{{p+10p}}}.



{{{(p^2+p)+(10p+10)}}} Group the terms into two pairs.



{{{p(p+1)+(10p+10)}}} Factor out the GCF {{{p}}} from the first group.



{{{p(p+1)+10(p+1)}}} Factor out {{{10}}} 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.



{{{(p+10)(p+1)}}} Combine like terms. Or factor out the common term {{{p+1}}}



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Answer:



So {{{p^2+11p+10}}} factors to {{{(p+10)(p+1)}}}.



In other words, {{{p^2+11p+10=(p+10)(p+1)}}}.



Note: you can check the answer by expanding {{{(p+10)(p+1)}}} to get {{{p^2+11p+10}}} or by graphing the original expression and the answer (the two graphs should be identical).


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