Introduction

Learning how to factor the quadratic polynomial is the first step to studying many other factoring techniques. This is why we will thoroughly review this technique and expand its use as far as possible. By the end, you will not only have learned how to factor the quadratic polynomial (of degree 2) but also use these same techniques to factor any (2n)-quadratic polynomial.

Quadratic and (2n)-Quadratic Polynomial

A quadratic polynomial is a polynomial of degree two. If you don’t know what the degree of a polynomial is, follow this link. Therefore, a quadratic polynomial is any function of the form

P(x) = ax^{2}+bx +c

with a,b,c\in\mathbb{R} and a\neq 0. However, our study will not only focus on factoring polynomials of this form but will aim at a generalized form of which the quadratic is just a particular case. We are talking about the (2n)-quadratic polynomial. This generalization encompasses all polynomials that can be written as

P(x) = ax^{2n}+bx^n +c

where, in addition to assuming a,b,c\in\mathbb{R} and a\neq 0, n\in\mathbb{N} is arbitrary. Examples of this type of polynomial include:

• P(x) = 3x^2 -x + 1
• Q(x) = 7x^4 +5x^2 + 3
• R(x) = -4x^6 +12x^3 + 2
• S(x) = 21x^8 -75 x^4 -9

and so on.

Factoring the quadratic polynomial

As we have seen, a degree 2 polynomial has the general form

P(x) = ax^{2}+bx +c \;\; , \;\; a\neq 0

Factoring is the process that breaks a complex polynomial into the product of two simpler polynomials. Thus, if factoring is possible, there are constants \alpha,\beta,\gamma,\delta \in\mathbb{R}, with \alpha, \gamma \neq 0 such that:

Since we have equality between the left and right sides, when one side cancels out, the other must also cancel out. It turns out that the right side cancels out when x=-\beta/\alpha or when x=-\delta/\gamma. Let’s now see for which values the left side of this equality cancels. We will have

From this reasoning, the Greek letter constants in the factorization must (without loss of generality) satisfy the following conditions:

• \alpha\gamma = a
• \displaystyle \frac{\beta}{\alpha} = - \left(\frac{-b + \sqrt{b^2 - 4ac }}{2a} \right)
• \displaystyle \frac{\delta}{\gamma} = - \left(\frac{-b - \sqrt{b^2 - 4ac }}{2a} \right)

And with this, we have a technique that will allow us to factor any degree 2 polynomial, and if it cannot be factored, it will notify you through the number under the square root: if that number is negative, then it cannot be factored (with real numbers). We can simplify all this by introducing the notation convention:

• x_1 =\displaystyle \frac{-b + \sqrt{b^2 - 4ac }}{2a}
• x_2 =\displaystyle \frac{-b - \sqrt{b^2 - 4ac }}{2a}

Which, in turn, is summarized in the good old

\color{blue}{x_{1,2} = \displaystyle \frac{-b \pm \sqrt{b^2 - 4ac }}{2a}} ✅

So the factorization will finally be in the form

\color{blue}{P(x) = ax^2 +bx + c = a(x-x_1)(x - x_2)}✅

Expanding to the factorization of the bi-quadratic polynomial

This technique can also be used to factor the bi-quadratic polynomial as follows:

Q(x) = ax^4 + bx^2 + c = a(x^2)^2 + bx^2 + c =a (x^2 - x_1^2)(x^2-x_2^2)

Where x^2_{1,2} = \displaystyle \dfrac{-b \pm \sqrt{b^2 - 4ac }}{2a}. So now you can write

Q(x) = ax^4 + bx^2 + c = a\left(x^2 - \displaystyle \dfrac{-b + \sqrt{b^2 - 4ac }}{2a}\right) \left(x^2- \dfrac{-b - \sqrt{b^2 - 4ac }}{2a}\right)

At this point, you must be careful, as what comes next has its restrictions. If x_1^2 is not a positive number, then you can use sum by difference to separate (x^2 - x_1^2) = (x-x_1)(x + x_1); otherwise, you will encounter complex numbers and therefore will no longer be able to factor in the reals. If the roots are all well-defined, you will be able to write:

\begin{array}{rl} Q(x) &= ax^4 + bx^2 + c \\ \\ & = a \left(x -\displaystyle \sqrt{\frac{-b + \sqrt{b^2 - 4ac }}{2a}}\right) \left(x + \displaystyle \sqrt{\frac{-b + \sqrt{b^2 - 4ac }}{2a}}\right) \\ \\ & \left(x- \displaystyle \sqrt{\frac{-b - \sqrt{b^2 - 4ac }}{2a}}\right) \left(x+ \sqrt{\displaystyle \frac{-b - \sqrt{b^2 - 4ac }}{2a}}\right) \end{array}

Otherwise, you will stop at the previous step.

Generalizing to the factorization of the (2n)-quadratic polynomial

With this, it is clear where the method is headed, to factor the (2n)-quadratic polynomial just rethink the way it is written and use the previous methods wherever the roots are well defined. In this way, we will have:

R(x) = a(x^n)^{2}+b (x^n) +c = a(x^n-x_1^n)(x^n-x_2^n)

Where x^n_{1,2} =\displaystyle \frac{-b \pm \sqrt{b^2 - 4ac }}{2a}. Then with this, we separate by sum by difference wherever complex numbers do not appear.

Example Exercises:

Now it’s your turn to try these techniques with some exercises. The polynomials below are chosen completely at random, so they will help you recognize the possible difficulties that can be encountered when factoring these things.

First Round

These polynomials are the ones I gave as examples at the beginning of this entry

1. P(x) = 3x^2 -x + 1
2. Q(x) = 7x^4 +5x^2 + 3
3. R(x) = -4x^6 +12x^3 + 2
4. S(x) = 21x^8 -75 x^4 -9

Second Round

And these are some other slightly more difficult ones.

1. P(x) = 78x^2 -21x - 13
2. Q(x) = 27x^4 +5x^2 - 14
3. R(x) = 9x^6 +12x^3 - 16
4. S(x) = -9x^8 -2 x^4 + 10
5. T(x) = 5x^{12} -2 x^6 - 15

Exercise Solutions