Prove that x-1x-1-ix-1-ix-1-i-x4-4

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Standard VIII

Mathematics

Expansion of (x+a)(x+b)

Prove that x+...

Question

Prove that (x+1)(x+1-i)(x-1+i)(x-1-i)=x⁴+4

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Solution

Taking LHS,

(x+1+i) (x+1-i) (x-1+i) (x-1-i)

= [(x+1+i) (x+1-i)] [ (x-1+i) (x-1-i)]

=[(x+1)^2 -i^2][(x-1)^2 -i^2]

=[x^2 +1+2x+1][x^2 -2x+1+1]

=[(x^2 +2)+2x][(x^2 +2)-2x]

=x^4 +4x^2 +4- 4x^2

=x^4 +4 = RHS

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Q.
The value of

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Prove that (x+1)(x+1-i)(x-1+i)(x-1-i)=x​​​​​​4+4

Taking LHS, (x+1+i) (x+1-i) (x-1+i) (x-1-i) = [(x+1+i) (x+1-i)] [ (x-1+i) (x-1-i)] =[(x+1)^2 -i^2][(x-1)^2 -i^2] =[x^2 +1+2x+1][x^2 -2x+1+1] =[(x^2 +2)+2x][(x^2 +2)-2x] =x^4 +4x^2 +4- 4x^2 =x^4 +4 = RHS

Prove that (x+1)(x+1-i)(x-1+i)(x-1-i)=x⁴+4

Taking LHS,

(x+1+i) (x+1-i) (x-1+i) (x-1-i)

= [(x+1+i) (x+1-i)] [ (x-1+i) (x-1-i)]

=[(x+1)^2 -i^2][(x-1)^2 -i^2]

=[x^2 +1+2x+1][x^2 -2x+1+1]

=[(x^2 +2)+2x][(x^2 +2)-2x]

=x^4 +4x^2 +4- 4x^2

=x^4 +4 = RHS