Respuesta :
Given expression : [tex]y^4+14y^2+49[/tex].
Soltution: Let us convert given expression in quadratic form.
y^4 =(y^2)^2.
Given expression could be written in quadratic form as,
(y2)^2 + 2 * (y^2) (7) +(7)^2.
If we compare that above expression by square of the sum identity,
(a)^2 + 2(a)(b) +(b)^2 = (a+b)^2.
We can see that a = y^2, b=7.
Substituting values of a and b in (a+b)^2, we get
[tex](y^2+7)^2[/tex]
Expanding above expression into two factors, we get
[tex](y^2+7)(y^2+7).[/tex]
[tex]\sqrt{-7} *\sqrt{-7}=i\sqrt{7}*i\sqrt{7}=(i\sqrt{7})^2[/tex]
But [tex](i\sqrt{7})^2=-7.[/tex].
We have +7 there.
So, 7=[tex]-(i\sqrt{7})^2[/tex][/tex]
Substituting this value of +7 in [tex](y^2+7)(y^2+7)[/tex] expression, we get
[tex](y^2+7)(y^2+7)=(y^2-(i\sqrt{7})^2)(y^2-(i\sqrt{7})^2)[/tex]
Applying difference of the squares identity (n)^2 -(m)^2 = (n-m)(n+m).
[tex](y^2-(i\sqrt{7})^2)=(y-i\sqrt{7})(y+i\sqrt{7})[/tex]
Therefore, final factored expression is
[tex](y^2-(i\sqrt{7})^2)(y^2-(i\sqrt{7})^2)=(y-i\sqrt{7})(y+i\sqrt{7})(y-i\sqrt{7})(y+i\sqrt{7})[/tex]
