Prove that $x^{4} + 4$ = $(x + 1 + i) (x + 1 - i) (x - 1 + i) (x - 1 - i) .$

Open in App

Solution

$(x + 1 + i) (x + 1 - i) (x - 1 + i) (x - 1 - i)$
= $[(x + 1)^{2} - i^{2}] [(x - 1)^{2} - i^{2}]$
= $(x^{2} + 2 x + 1 + 1) (x^{2} - 2 x + 1 + 1)$
= $[(x^{2} + 2) + 2 x] [(x^{2} + 2) - 2 x]$
= $(x^{2} + 2)^{2} - 4 x^{2} = x^{4} + 4 x^{2} + 4 - 4 x^{2} = x^{4} + 4$

Suggest Corrections

Similar questions

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

Q. If

A = [\begin{matrix} \frac{1}{2} (e^{i x} + e^{- i x}) & \frac{1}{2} (e^{i x} - e^{- i x}) \frac{1}{2} (e^{i x} - e^{- i x}) & \frac{1}{2} (e^{i x} + e^{- i x}) \end{matrix}]

then

A^{- 1}

exists

The inflation rate of a country follows a certain trend given by $I = I_{X} + I_{X - 1} + I_{X - 2}$ where $[I_{X}] = 0, [I_{X - 1}] = 1$ and $[I_{X - 2}] = 2$ , [ ] represents greatest integer function. Find the range of the inflation rate.

Q. No.of nonzero terms in

(1 + x)^{42} + (1 - x)^{42} + (1 + i x)^{42} + (1 - i x)^{42}

\frac{(1 + i) x - i}{2 + i} + \frac{(1 + 2 i) y + i}{2 - i} = 1 \Rightarrow (x, y) =

Join BYJU'S Learning Program

Prove that x4+4 = (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−i2][(x−1)2−i2]= (x2 +2x+1+1)(x2−2x+1+1)= [(x2 +2)+2x][(x2+2)−2x]= (x2+2)2−4x2=x4+4x2+4−4x2=x4+4

Prove that $x^{4} + 4$ = $(x + 1 + i) (x + 1 - i) (x - 1 + i) (x - 1 - i) .$

$(x + 1 + i) (x + 1 - i) (x - 1 + i) (x - 1 - i)$
= $[(x + 1)^{2} - i^{2}] [(x - 1)^{2} - i^{2}]$
= $(x^{2} + 2 x + 1 + 1) (x^{2} - 2 x + 1 + 1)$
= $[(x^{2} + 2) + 2 x] [(x^{2} + 2) - 2 x]$
= $(x^{2} + 2)^{2} - 4 x^{2} = x^{4} + 4 x^{2} + 4 - 4 x^{2} = x^{4} + 4$