Respuesta :
[tex]x^4-625=(x^2-25)(x^2+25)=(x-5)(x+5)(x-5i)(x+5i)\\\\x^4+10x^2+25=(x^2+5)^2=((x-\sqrt5i)(x+\sqrt5i))^2=(x-\sqrt5i)^2(x+\sqrt5i)^2[/tex]
Answer:
1) [tex]x^4-625=(x+5i)(x-5i)(x+5)(x-5)[/tex]
2) [tex]x^4+10x^2+25=(x+\sqrt{5}i)^2(x-\sqrt5 i)^2[/tex]
Step-by-step explanation:
1) Given : Expression [tex]x^4-625[/tex]
To find : Factor the expression completely over the complex numbers ?
Solution :
We can re-write the expression as,
[tex]x^4-625=(x^2)^2-(25)^2[/tex]
Applying identity, [tex]a^2-b^2 = (a+b)(a-b)[/tex]
[tex]x^4-625=(x^2+25)(x^2-25)[/tex]
[tex]x^4-625=(x^2+25)(x^2-5^2)[/tex]
Again apply same identity,
[tex]x^4-625=(x^2+25)(x+5)(x-5) [/tex]
The factor of [tex]x^2+25=(x+5i)(x-5i)[/tex]
Factor form is [tex]x^4-625=(x+5i)(x-5i)(x+5)(x-5)[/tex]
2) Given : Expression [tex]x^4+10x^2+25[/tex]
To find : Factor the expression completely over the complex numbers ?
Solution :
Expression [tex]x^4+10x^2+25[/tex]
Let [tex]x^2=y[/tex]
[tex]y^2+10y+25[/tex]
To factor we equate it to zero.
[tex]y^2+10y+25=0[/tex]
Apply middle term split,
[tex]y^2+5y+5y+25=0[/tex]
[tex]y(y+5)+5(y+5)=0[/tex]
[tex](y+5)(y+5)=0[/tex]
Substitute back,
[tex](x^2+5)(x^2+5)=0[/tex]
[tex](x^2+5)^2=0[/tex]
[tex]x^2+5=0[/tex]
[tex]x^2=-5[/tex]
Taking root both side,
[tex]x=\sqrt{-5}[/tex]
[tex]x=\pm \sqrt{5}i[/tex]
So, The factors are [tex](x+\sqrt{5}i)(x-\sqrt5 i)[/tex]
Factor form is [tex]x^4+10x^2+25=(x+\sqrt{5}i)^2(x-\sqrt5 i)^2[/tex]
