Question

Answer 1

a: \(x^4+4=x^4+4x^2+4-4x^2\)

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

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

b: \(4x^8+1\)

\(=4x^8+4x^4+1-4x^4\)

\(=\left(2x^4+1\right)^2-\left(2x^2\right)^2\)

\(=\left(2x^4-2x^2+1\right)\left(2x^4+2x^2+1\right)\)

c: \(x^4+5x^2+9\)

\(=x^4+6x^2+9-x^2\)

\(=\left(x^2+3\right)^2-x^2=\left(x^2+3-x\right)\left(x^2+3+x\right)\)

Answer 2

`a,` `x^4 + 4`

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

`=(x^2+2)^2 - (2x)^2`

`=(x^2 -2x+2)(x^2 + 2x +2)`

`b,` `4x ^ 8 + 1`

`= 4x ^ 8 + 4x ^ 4 + 1 - 4x ^ 4`

`= (2x ^ 4 + 1) ^ 2 - (2x ^ 2) ^ 2`

`= (2x ^ 4 - 2x ^ 2 + 1)(2x ^ 4 + 2x ^ 2 + 1)`

`c,` `x ^ 4 + 5x ^ 2 + 9`

`= x ^ 4 + 6x ^ 2 + 9 - x ^ 2`

`= (x ^ 2 + 3) ^ 2 - x ^ 2`

`= (x ^ 2 + 3 - x)(x ^ 2 + 3 + x)`

Answer 3

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

\(4x^8+1=4x^8+4x^4+1-4x^4=\left(2x^4+1\right)^2-\left(2x^2\right)^2=\left(2x^4-2x^2+1\right)\left(2x^4+2x^2+1\right)\)

\(x^4+5x^2+9=x^4+6x^2+9-x^2=\left(x^2+3\right)^2-x^2=\left(x^2-x+3\right)\left(x^2+x+3\right)\)