<PRE><B>Question 191477</B>


{{{70a^2b+27ab-9b}}} Start with the given expression



{{{b(70a^2+27a-9)}}} Factor out the GCF {{{b}}}



Now let&#39;s focus on the inner expression {{{70a^2+27a-9}}}





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Looking at {{{70a^2+27a-9}}} we can see that the first term is {{{70a^2}}} and the last term is {{{-9}}} where the coefficients are 70 and -9 respectively.


Now multiply the first coefficient 70 and the last coefficient -9 to get -630. Now what two numbers multiply to -630 and add to the  middle coefficient 27? Let&#39;s list all of the factors of -630:




Factors of -630:

1,2,3,5,6,7,9,10,14,15,18,21,30,35,42,45,63,70,90,105,126,210,315,630


-1,-2,-3,-5,-6,-7,-9,-10,-14,-15,-18,-21,-30,-35,-42,-45,-63,-70,-90,-105,-126,-210,-315,-630 ...List the negative factors as well. This will allow us to find all possible combinations


These factors pair up and multiply to -630

(1)*(-630)

(2)*(-315)

(3)*(-210)

(5)*(-126)

(6)*(-105)

(7)*(-90)

(9)*(-70)

(10)*(-63)

(14)*(-45)

(15)*(-42)

(18)*(-35)

(21)*(-30)

(-1)*(630)

(-2)*(315)

(-3)*(210)

(-5)*(126)

(-6)*(105)

(-7)*(90)

(-9)*(70)

(-10)*(63)

(-14)*(45)

(-15)*(42)

(-18)*(35)

(-21)*(30)


note: remember, the product of a negative and a positive number is a negative number



Now which of these pairs add to 27? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 27


&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;1&lt;/td&gt;&lt;td align=&quot;center&quot;&gt;-630&lt;/td&gt;&lt;td&gt;1+(-630)=-629&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td align=&quot;center&quot;&gt;2&lt;/td&gt;&lt;td align=&quot;center&quot;&gt;-315&lt;/td&gt;&lt;td&gt;2+(-315)=-313&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td align=&quot;center&quot;&gt;3&lt;/td&gt;&lt;td align=&quot;center&quot;&gt;-210&lt;/td&gt;&lt;td&gt;3+(-210)=-207&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td align=&quot;center&quot;&gt;5&lt;/td&gt;&lt;td align=&quot;center&quot;&gt;-126&lt;/td&gt;&lt;td&gt;5+(-126)=-121&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td align=&quot;center&quot;&gt;6&lt;/td&gt;&lt;td align=&quot;center&quot;&gt;-105&lt;/td&gt;&lt;td&gt;6+(-105)=-99&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td align=&quot;center&quot;&gt;7&lt;/td&gt;&lt;td align=&quot;center&quot;&gt;-90&lt;/td&gt;&lt;td&gt;7+(-90)=-83&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td align=&quot;center&quot;&gt;9&lt;/td&gt;&lt;td align=&quot;center&quot;&gt;-70&lt;/td&gt;&lt;td&gt;9+(-70)=-61&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td align=&quot;center&quot;&gt;10&lt;/td&gt;&lt;td align=&quot;center&quot;&gt;-63&lt;/td&gt;&lt;td&gt;10+(-63)=-53&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td align=&quot;center&quot;&gt;14&lt;/td&gt;&lt;td align=&quot;center&quot;&gt;-45&lt;/td&gt;&lt;td&gt;14+(-45)=-31&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td align=&quot;center&quot;&gt;15&lt;/td&gt;&lt;td align=&quot;center&quot;&gt;-42&lt;/td&gt;&lt;td&gt;15+(-42)=-27&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td align=&quot;center&quot;&gt;18&lt;/td&gt;&lt;td align=&quot;center&quot;&gt;-35&lt;/td&gt;&lt;td&gt;18+(-35)=-17&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td align=&quot;center&quot;&gt;21&lt;/td&gt;&lt;td align=&quot;center&quot;&gt;-30&lt;/td&gt;&lt;td&gt;21+(-30)=-9&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td align=&quot;center&quot;&gt;-1&lt;/td&gt;&lt;td align=&quot;center&quot;&gt;630&lt;/td&gt;&lt;td&gt;-1+630=629&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td align=&quot;center&quot;&gt;-2&lt;/td&gt;&lt;td align=&quot;center&quot;&gt;315&lt;/td&gt;&lt;td&gt;-2+315=313&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td align=&quot;center&quot;&gt;-3&lt;/td&gt;&lt;td align=&quot;center&quot;&gt;210&lt;/td&gt;&lt;td&gt;-3+210=207&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td align=&quot;center&quot;&gt;-5&lt;/td&gt;&lt;td align=&quot;center&quot;&gt;126&lt;/td&gt;&lt;td&gt;-5+126=121&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td align=&quot;center&quot;&gt;-6&lt;/td&gt;&lt;td align=&quot;center&quot;&gt;105&lt;/td&gt;&lt;td&gt;-6+105=99&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td align=&quot;center&quot;&gt;-7&lt;/td&gt;&lt;td align=&quot;center&quot;&gt;90&lt;/td&gt;&lt;td&gt;-7+90=83&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td align=&quot;center&quot;&gt;-9&lt;/td&gt;&lt;td align=&quot;center&quot;&gt;70&lt;/td&gt;&lt;td&gt;-9+70=61&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td align=&quot;center&quot;&gt;-10&lt;/td&gt;&lt;td align=&quot;center&quot;&gt;63&lt;/td&gt;&lt;td&gt;-10+63=53&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td align=&quot;center&quot;&gt;-14&lt;/td&gt;&lt;td align=&quot;center&quot;&gt;45&lt;/td&gt;&lt;td&gt;-14+45=31&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td align=&quot;center&quot;&gt;-15&lt;/td&gt;&lt;td align=&quot;center&quot;&gt;42&lt;/td&gt;&lt;td&gt;-15+42=27&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td align=&quot;center&quot;&gt;-18&lt;/td&gt;&lt;td align=&quot;center&quot;&gt;35&lt;/td&gt;&lt;td&gt;-18+35=17&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td align=&quot;center&quot;&gt;-21&lt;/td&gt;&lt;td align=&quot;center&quot;&gt;30&lt;/td&gt;&lt;td&gt;-21+30=9&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;



From this list we can see that -15 and 42 add up to 27 and multiply to -630



Now looking at the expression {{{70a^2+27a-9}}}, replace {{{27a}}} with {{{-15a+42a}}} (notice {{{-15a+42a}}} adds up to {{{27a}}}. So it is equivalent to {{{27a}}})


{{{70a^2+highlight(-15a+42a)-9}}}



Now let&#39;s factor {{{70a^2-15a+42a-9}}} by grouping:



{{{(70a^2-15a)+(42a-9)}}} Group like terms



{{{5a(14a-3)+3(14a-3)}}} Factor out the GCF of {{{5a}}} out of the first group. Factor out the GCF of {{{3}}} out of the second group



{{{(5a+3)(14a-3)}}} Since we have a common term of {{{14a-3}}}, we can combine like terms


So {{{70a^2-15a+42a-9}}} factors to {{{(5a+3)(14a-3)}}}



So this also means that {{{70a^2+27a-9}}} factors to {{{(5a+3)(14a-3)}}} (since {{{70a^2+27a-9}}} is equivalent to {{{70a^2-15a+42a-9}}})




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So our expression goes from {{{b(70a^2+27a-9)}}} and factors further to {{{b(5a+3)(14a-3)}}}



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Answer:


So {{{70a^2b+27ab-9b}}} completely factors to {{{b(5a+3)(14a-3)}}}

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