factor the polynomial as a product of linear and a factor of a constant or an irreducible polynimial
6X^3-5X^2+3X-1
Thanks guys
Hello, victorfk06!
Are you familiar with the Factor Theorem?Factor the polynomial: .$\displaystyle 6x^3-5x^2+3x-1$
Given a polynomial $\displaystyle p(x)$, if $\displaystyle p(a) = 0$, then $\displaystyle (x-a)$ is a factor of $\displaystyle p(x).$
And the Rational Roots Theorem?
A rational root of a polynomial equation is of the form: .$\displaystyle \frac{n}{d}$
. . where $\displaystyle n$ is a factor of the constant term,
. . and $\displaystyle d$ is a factor of the leading coefficient.
We want a number which makes the polynomial equal zero.
There are eight candidates.
. . The numerator is a factor of 1: .$\displaystyle \pm1$
. . The denominator is a factor of 6: .$\displaystyle \pm1,\:\pm2,\:\pm3,\:\pm6$
The possible rational roots are: .$\displaystyle \pm1,\;\pm\tfrac{1}{2},\:\pm\tfrac{1}{3},\:\pm\tfr ac{1}{6}$
And we find that $\displaystyle x = \tfrac{1}{2}$ works: .$\displaystyle 6\left(\tfrac{1}{2}\right)^3 - 5\left(\tfrac{1}{2}\right)^2 + 3\left(\tfrac{1}{2}\right) - 1 \;=\;0$
Hence, $\displaystyle \left(x-\tfrac{1}{2}\right)\;\hdots\;\text{ or }(2x-1)$ is a factor.
Using long division, we find that:
. . $\displaystyle 6x^3 - 5x^2 + 3x - 1 \;=\;(2x-1)(3x^2-x+1)$ . . . as emathinstruction already pointed out.
We find that the quadratic expression doesn't factor . . . so we're done!