Factoring a Polynomial, write the polynomial (a) as the...

Factoring a Polynomial, write the polynomial (a) as the product of factors that are irreducible over the rationals, (b) as the product of linear and quadratic factors that are irreducible over the reals, and (c) in completely factored form. $$\begin{array}{l}{f(x)=x^{4}-2 x^{3}-3 x^{2}+12 x-18} \\ {\left(\text {Hint} : \text { One factor is } x^{2}-6 .\right)}\end{array}$$

Factoring a Polynomial, write the polynomial (a) as the product of factors that are irreducible over the rationals, (b) as the product of linear and quadratic factors that are irreducible over the reals, and (c) in completely factored form.
$$\begin{array}{l}{f(x)=x^{4}-2 x^{3}-3 x^{2}+12 x-18} \\ {\left(\text {Hint} : \text { One factor is } x^{2}-6 .\right)}\end{array}$$

Key Concepts

Complete Factorization (over the Complex Numbers)

Complete factorization refers to expressing a polynomial as a product of linear factors over the complex numbers. This method is supported by the Fundamental Theorem of Algebra, which guarantees that every non-constant polynomial has at least one complex root, ensuring that all polynomial factors can eventually be expressed in a linear form once complex roots are admitted.

Factoring Over the Reals

When factoring a polynomial over the reals, the polynomial is expressed as a product of linear factors corresponding to real roots and quadratic factors that are irreducible over the reals. Quadratic irreducible factors emerge when the polynomial has complex roots, which occur in conjugate pairs, but cannot be factored further using real numbers.

Polynomial Factorization

This is the process of breaking down a polynomial into a product of simpler polynomials (factors), which when multiplied together give back the original polynomial. The goal is to express a polynomial in a form that reveals its structures, such as roots and multiplicities, making it easier to solve equations, simplify expressions, or analyze the behavior of the polynomial.

Irreducibility over the Rationals

A polynomial factor is considered irreducible over the rationals if it cannot be factored further into polynomials with rational coefficients. This concept is essential when expressing a polynomial as a product of factors that have no non-trivial factors in the rational number system, thus providing a simplified yet complete factorization over the rationals.

Transcript

00:01 We're given a polynomial, f of x equals x to the fourth minus 2x cubed minus 3x squared plus 12x minus 18.

00:20 Now in part a, we're asked to write this polynomial as a product of factors that are irreducible over the rational.

00:27 We're given a hint that one factor is x squared minus 6.

00:31 So to apply this hint, i'll use long division.

00:36 So we're going to divide our polynomial.

00:39 X to the 4th minus 2x cubed minus 3x squared plus 12x minus 18 divided by x squared minus 6.

00:57 Well, this fits into x to the 4th x squared times.

01:00 Times, so we have x squared times this is x to the fourth minus 6x squared, and so i subtract and i get negative 2 minus negative 6, positive, this is negative 2x cubed i should say.

01:27 This is negative 2x cubed.

01:30 Then we have negative 3 minus negative 6 is plus 3x squared.

01:36 Now x squared minus 6 fits into this.

01:41 Negative 2x times.

01:44 And so we have negative 2x cubed plus 12x.

02:04 Subtracting, i get 3x squared, and then i pull down a plus 12x, and we subtract 12x from this, we just have 3x squared left over.

02:18 And then x squared minus 6 fits into this three times.

02:31 So this is 3x squared minus 18.

02:37 I'll pull down negative 18.

02:43 And then 3x squared minus 3x squared is 0.

02:47 And negative 18 minus negative 18 is 0.

02:51 The remainder is just 0...