Section 001 : Factoring quartic equation Special case

Introduction 

This section describes factoring a quartic polynomial with real co-efficients.
A polynomial of  $4^{th}$ degree is called a quartic polynomial. This if of the form

$P(x) = a_4x^4+a_3x^3+a_2x^2+a_1x+ a_0$

In general it is quite difficult to factor a quartic polynomial. But in some cases it can be done
as below in case this fits the following form that is difference of 2 squares

In case $a_3$ and $a_1$ are zero and $a_4$ and $a_0$ are perfect square in some cases we can add $x^2$ term to make it
a perfect square to make it a perfect square this becomes difference of 2 perfect squares and can be factored.
there shall be 2 choices coefficients for $x^2$ one positive and another negative.
We show the same below with example

Problem 1

Factor  $y^4-y^2+16$

Solution 1

This can be factored by completing the square. By changing the $y^2$ term
$y^4 -y^2 +16$
= $y^4 -y^2+8y^2-8y^2 +16$
= $y^4 +8y^2 +16 -9y^2$
= $(y^2+4)^2 -(3y)^2$
= $(y^2-3y+4)(y^2+3y+4)$