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
$y^4 -y^2 +16$
= $y^4 -y^2+8y^2-8y^2 +16$
= $y^4 +8y^2 +16 -9y^2$
= $(y^2+4)^2 -(3y)^2$
= $(y^2-3y+4)(y^2+3y+4)$

Welcome to my blog on math notes

You are welcome to my blog of math notes. In this blog I shall be providing some discussion of various notes of maths and how to apply the methods. this shall be mentioning the theory and application of the theory at school level maths. I shall provide the following topics. If you  are interested in some more topics then kindly add as a comment and I shall try to provide theory on that.

The following topics shall be covered and when some topic is covered the link to the topic shall be provided

1) Principle of Mathematical Induction
2) Extended Euclidean Algorithm
3) Chinese remainder theorem
4) polynomials
5) Multinomials
6) Inequality
7) LCM/GCD
8) permutation and combination
9) maximum/minimum
10)FLT
11)Prime factors
12)pigeon hole principle
13) Difference Equation
14) Binomial Theorem
15) Ratio and Proportion