Polynomial analysis into multiplication a) x4 + 4 b) 4x4 + 81 c) 4x4 - 9x2 + 81

Question by Kaya Renger

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24/09/2017 at 10:03

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Polynomial analysis into multiplication

a) x⁴ + 4

b) 4x⁴ + 81

c) 4x⁴ - 9x² + 81

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Dao Trong Luan 24/09/2017 at 10:55

a,

$x^4+4$

$=\left[\left(x^2\right)^2+4x^2+4\right]-4x^2$

$=\left[\left(x^2\right)^2+2\cdot x^2\cdot2+2^2\right]-\left(2x\right)^2$

$=\left(x^2+2\right)^2-\left(2x\right)^2$

$=\left(x^2+2+2x\right)\left(x^2+2-2x\right)$ Apply $a^2-b^2=\left(a+b\right)\left(a-b\right)$

b,

$4x^2+81$

$=2^2\cdot\left(x^2\right)^2+81$

$=\left[\left(2x^2\right)^2+36x^2+81\right]-36x^2$

$=\left[\left(2x^2\right)^2+2\cdot2x^2\cdot9+9^2\right]-\left(6x\right)^2$

$=\left(2x^2+9\right)^2-\left(6x\right)^2$

$=\left(2x^2+9+6x\right)\left(2x^2+9-6x\right)$

I'm in 7th grade so I can not do sentence c
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Vũ Trung Dũng 24/09/2017 at 14:15

a) $x^{4} + 4 = x^{4} + 4 x^{2} + 4 - 4 x^{2} = {(x^{2} + 2)}^{2} - {(2 x)}^{2}$

$= (x^{2} - 2 x + 2) (x^{2} + 2 x + 2)$

b) $4 x^{4} + 81 = 4 x^{4} + 36 x^{2} + 81 - 36 x^{2} = {(2 x^{2} + 9)}^{2} - {(6 x)}^{2}$

$= (2 x^{2} - 6 x + 9) (2 x^{2} + 6 x + 9)$

c) $4 x^{4} - 9 x^{2} + 81 = 4 x^{4} + 36 x^{2} + 81 - 45 x^{2}$

$= {(2 x^{2} + 9)}^{2} - {(\sqrt{45} x)}^{2} = (2 x^{2} - \sqrt{45} x + 9) (2 x^{2} + \sqrt{45} x + 9)$
Phan Thanh Tinh Coordinator 24/09/2017 at 11:11

a) $x^4+4=x^4+4x^2+4-4x^2=\left(x^2+2\right)^2-\left(2x\right)^2$

$=\left(x^2-2x+2\right)\left(x^2+2x+2\right)$

b) $4x^4+81=4x^4+36x^2+81-36x^2=\left(2x^2+9\right)^2-\left(6x\right)^2$

$=\left(2x^2-6x+9\right)\left(2x^2+6x+9\right)$

c) $4x^4-9x^2+81=4x^4+36x^2+81-45x^2$

$=\left(2x^2+9\right)^2-\left(\sqrt{45}x\right)^2=\left(2x^2-\sqrt{45}x+9\right)\left(2x^2+\sqrt{45}x+9\right)$

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