Factoring two cubic polynomials with rational roots

[NotaMathPerson]

After successfully solving tons of problems about factoring polynomials I've buit up quite a confidence in myself. But it did not last long when I stumbled upon these problems

Factor
$3x^3-13x^2+23x-21$
$ 6x^3-2x^2-13x-6$

None of the techniques I knew were able to help me solve these problems. Kindly assist me in this matter. Thanks much!

 

[MarkFL]

Hello and welcome to MHB, NotaMathPerson! (Wave)

Let's walk through the first expression, and then you can try the second. Let's let:

$$f(x)=3x^3-13x^2+23x-21$$

Using the Rational Root Theorem, by trial and error, we find:

$$f\left(\frac{7}{3}\right)=0$$

So, using synthetic division, we find:

$$\begin{array}{c|rr}& 3 & -13 & 23 & -21 \\ \frac{7}{3} & & 7 & -14 & 21 \\ \hline & 3 & -6 & 9 & 0 \end{array}$$

So, we may now state:

$$f(x)=3x^3-13x^2+23x-21=\left(x-\frac{7}{3}\right)\left(3x^2-6x+9\right)=(3x-7)\left(x^2-2x+3\right)$$