factorize-x-1-x-3-x-5-x-7-16-

Question Number 64508 by lalitchand last updated on 18/Jul/19

factorize (x + 1) (x + 3) (x + 5) (x + 7) + 16

Commented by Prithwish sen last updated on 18/Jul/19

(x + 1) (x + 7) (x + 3) (x + 5) + 16

= (x^{2} + 8 x + 7) (x^{2} + 8 x + 15) + 16

= (a + 7) (a + 15) + 16 putting x^{2} + 8 x = a

= a^{2} + 22 a + 121

= {(a + 11)}^{2}

= (x^{2} + 8 x + 11) (x^{2} + 8 x + 11)

Answered by behi83417@gmail.com last updated on 18/Jul/19

(x + 1) (x + 7) = x^{2} + 8 x + 7

(x + 3) (x + 5) = x^{2} + 8 x + 15

x^{2} + 8 x = p \Rightarrow (p + 7) (p + 15) + 16 =

= p^{2} + 22 p + 121 = {(p + 11)}^{2}

\Rightarrow (x + 1) (x + 3) (x + 5) (x + 7) + 16 =

= {(x^{2} + 8 x + 11)}^{2} .

Answered by MJS last updated on 18/Jul/19

put x = t - 4

(t - 3) (t - 1) (t + 1) (t + 3) + 16

t^{4} - 10 t^{2} + 25

{(t^{2} - 5)}^{2}

put t = x + 4

{({(x + 4)}^{2} - 5)}^{2}

{(x + 4 - \sqrt{5})}^{2} {(x + 4 + \sqrt{5})}^{2}

or

{(x^{2} + 8 x + 11)}^{2}

Commented by Prithwish sen last updated on 18/Jul/19

great idea sir .