factorise question 33 35 37
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33: we need to factorize.
(x-1)(x-3)(x-5)(x-7)-64
we can buy the rational root theorem, see that there are no root of this polynomial. Next observation is that if the first part of the polynomial is a square we can rewrite the whole polynomial as the difference of two squares. But it turns out that the first part of the polynomial is not a square. However, we can note that
(x-1)(x-7)=(x2)-8x+7
(x-1)(x-7)=(x2)-8x+15
(x-3)(x-5)=(x2)-8x+15
therefore, letting
(x2)-8x+7=p(x2)-8x+7=p, we can rewrite the given polynomial as
p(p+8)-64=p 2+8p-64
now it is the the factorisable
hope it helps
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