<PRE><B>Question 610313</B>
Looking at the expression {{{64a^2-16ab+b^2}}}, we can see that the first coefficient is {{{64}}}, the second coefficient is {{{-16}}}, and the last coefficient is {{{1}}}.



Now multiply the first coefficient {{{64}}} by the last coefficient {{{1}}} to get {{{(64)(1)=64}}}.



Now the question is: what two whole numbers multiply to {{{64}}} (the previous product) &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to the second coefficient {{{-16}}}?



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 {{{64}}} (the previous product).



Factors of {{{64}}}:

1,2,4,8,16,32,64

-1,-2,-4,-8,-16,-32,-64



Note: list the negative of each factor. This will allow us to find all possible combinations.



These factors pair up and multiply to {{{64}}}.

1*64 = 64
2*32 = 64
4*16 = 64
8*8 = 64
(-1)*(-64) = 64
(-2)*(-32) = 64
(-4)*(-16) = 64
(-8)*(-8) = 64


Now let&#39;s add up each pair of factors to see if one pair adds to the middle coefficient {{{-16}}}:



&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;64&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;1+64=65&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;2&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;32&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;2+32=34&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;4&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;16&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;4+16=20&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;8&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;8&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;8+8=16&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;-64&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-1+(-64)=-65&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-2&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-32&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-2+(-32)=-34&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-4&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-16&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-4+(-16)=-20&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;-8&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;-8&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;-8+(-8)=-16&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;



From the table, we can see that the two numbers {{{-8}}} and {{{-8}}} add to {{{-16}}} (the middle coefficient).



So the two numbers {{{-8}}} and {{{-8}}} both multiply to {{{64}}} &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to {{{-16}}}



Now replace the middle term {{{-16ab}}} with {{{-8ab-8ab}}}. Remember, {{{-8}}} and {{{-8}}} add to {{{-16}}}. So this shows us that {{{-8ab-8ab=-16ab}}}.



{{{64a^2+highlight(-8ab-8ab)+b^2}}} Replace the second term {{{-16ab}}} with {{{-8ab-8ab}}}.



{{{(64a^2-8ab)+(-8ab+b^2)}}} Group the terms into two pairs.



{{{8a(8a-b)+(-8ab+b^2)}}} Factor out the GCF {{{8a}}} from the first group.



{{{8a(8a-b)-b(8a-b)}}} Factor out the {{{-b}}} from the second group.



{{{(8a-b)(8a-b)}}} Factor out {{{8a-b}}} from the entire expression.


===============================================================



Answer:



So {{{64a^2-16ab+b^2}}} factors to {{{(8a-b)(8a-b)}}}.



In other words, {{{64a^2-16ab+b^2=(8a-b)(8a-b)}}} for all values of &#39;a&#39; and &#39;b&#39;.



Note: you can check the answer by expanding {{{(8a-b)(8a-b)}}} to get {{{64a^2-16ab+b^2}}} back again or by graphing the original expression and the answer (the two graphs should be identical).
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