<PRE><B>Question 473308</B>


Looking at the expression {{{25r^2+10r+1}}}, we can see that the first coefficient is {{{25}}}, the second coefficient is {{{10}}}, and the last term is {{{1}}}.



Now multiply the first coefficient {{{25}}} by the last term {{{1}}} to get {{{(25)(1)=25}}}.



Now the question is: what two whole numbers multiply to {{{25}}} (the previous product) &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to the second coefficient {{{10}}}?



To find these two numbers, we need to list &lt;font size=4&gt;&lt;b&gt;all&lt;/b&gt;&lt;/font&gt; of the factors of {{{25}}} (the previous product).



Factors of {{{25}}}:

1,5,25

-1,-5,-25



Note: list the negative of each factor. This will allow us to find all possible combinations.



These factors pair up and multiply to {{{25}}}.

1*25 = 25
5*5 = 25
(-1)*(-25) = 25
(-5)*(-5) = 25


Now let&#39;s add up each pair of factors to see if one pair adds to the middle coefficient {{{10}}}:



&lt;table border=&quot;1&quot;&gt;&lt;th&gt;First Number&lt;/th&gt;&lt;th&gt;Second Number&lt;/th&gt;&lt;th&gt;Sum&lt;/th&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;1&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;25&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;1+25=26&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;5&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;5&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;5+5=10&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-1&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-25&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-1+(-25)=-26&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-5&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-5&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-5+(-5)=-10&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;



From the table, we can see that the two numbers {{{5}}} and {{{5}}} add to {{{10}}} (the middle coefficient).



So the two numbers {{{5}}} and {{{5}}} both multiply to {{{25}}} &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to {{{10}}}



Now replace the middle term {{{10r}}} with {{{5r+5r}}}. Remember, {{{5}}} and {{{5}}} add to {{{10}}}. So this shows us that {{{5r+5r=10r}}}.



{{{25r^2+highlight(5r+5r)+1}}} Replace the second term {{{10r}}} with {{{5r+5r}}}.



{{{(25r^2+5r)+(5r+1)}}} Group the terms into two pairs.



{{{5r(5r+1)+(5r+1)}}} Factor out the GCF {{{5r}}} from the first group.



{{{5r(5r+1)+1(5r+1)}}} Factor out {{{1}}} 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.



{{{(5r+1)(5r+1)}}} Combine like terms. Or factor out the common term {{{5r+1}}}



{{{(5r+1)^2}}} Condense the terms.



===============================================================



Answer:



So {{{25r^2+10r+1}}} factors to {{{(5r+1)^2}}}.



In other words, {{{25r^2+10r+1=(5r+1)^2}}}.



Note: you can check the answer by expanding {{{(5r+1)^2}}} to get {{{25r^2+10r+1}}} or by graphing the original expression and the answer (the two graphs should be identical).
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