SOLUTION: I need help factoring the two following expressions: {{{ 36(2a^2 - 3)^2 - 44(2a^2 - 3) - 15 }}} {{{ 81(7a+3)^4 - 72(7a+3)^2 + 16}}} Thanks!

Algebra ->  Finance -> SOLUTION: I need help factoring the two following expressions: {{{ 36(2a^2 - 3)^2 - 44(2a^2 - 3) - 15 }}} {{{ 81(7a+3)^4 - 72(7a+3)^2 + 16}}} Thanks!      Log On


   

Question 1200303: I need help factoring the two following expressions:
+36%282a%5E2+-+3%29%5E2+-+44%282a%5E2+-+3%29+-+15+
+81%287a%2B3%29%5E4+-+72%287a%2B3%29%5E2+%2B+16
Thanks!
Answer by ikleyn(52781) About Me  (Show Source):
You can put this solution on YOUR website!
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Consider  81*(7a+3)^4 - 72*(7a+3)^2 + 16.


To simplify writing, introduce new variable u = (7a+3)^2.


Then the given expression takes the form  81u^2 -72u + 16.


From the first glance, it is clear that this expression is a full square

    81u^2 -72u + 16 = (9u - 4)^2.


Now return (7a+3)^2 instead of u into this formula.  You will get

    81*(7a+3)^4 - 72*(7a+3)^2 + 16 = 81u^2 -72u + 16 = (9u - 4)^2 = (9*(7a+3)^2-4)^2 = 

    = ((21a+9)^2 - 4)^2.


Next, represent (21a+9)^2 - 4 as the difference of squares and factor it further

    (21a+9)^2 - 4 = (21a+9)^2 - 2^2 = ((21a+9)-2)*((21a+9)+2) = (21a+7)*(21a+11).


Therefore and finally,

    81*(7a+3)^4 - 72*(7a+3)^2 + 16 = (21a+7)^2*(21a+11)^2.    ANSWER


It is your desired factoring of the given original expression.

Solved.