SOLUTION: show that 3x-2 is a factor of 3x^2+x^2-20x+12. thank you.

Question 152146: show that 3x-2 is a factor of 3x^2+x^2-20x+12. thank you.
Answer by nabla(475) (Show Source): You can put this solution on YOUR website!
Note: I assume that the 3x^2 term should really be 3x^3.
Note 2: You could show that (3x-2) is a factor of (3x^3+x^2-20x+12) through long or synthetic division. Here I will show you my attempt at a proof.

I will prove this in the most elegant fashion I can.
Proof. We use a direct proof.
Predicate: (3x-2) is a factor of 3x^3+x^2-20x+12.
By definition, a cubic, if factorisable, will factor as (ax+b)(cx^2+dx+e), with {a, b, c, d, e} particular constants.
So for our predicate, in order for (3x-2) to be a factor, we need (3x-2)(cx^2+dx+e).
Moreover, (3x-2)(cx^2+dx+e)=3x^3+x^2-20x+12
and, -2e-2dx+3ex-2cx^2+3dx^2+3cx^3=3x^3+x^2-20x+12.
By definition, polynomials are equivalent if their variables' coefficients are equal.
Thus:
-2e=12 implies e=-6
-2d+3e=-20
-2c+3d=1
3c=3 implies c=1
And it follows that:
-2d-18=-20 implies d=1.
This gives a factorization of (3x-2)(x^2+x-6)=(3x-2)(x-2)(x+3). Thus we are certain that (3x-2) is a factor of polynomial (3x^3+x^2-20x+12). QED
RELATED QUESTIONS
Show that(3x-2) is a factor of... (answered by josgarithmetic)

3x^2-20x-7+... (answered by scott8148)

Given that {{{ x^2-3x+2 }}} is a factor of {{{ x^4+kx^3-10x^2-20x+24 }}}, evaluate the... (answered by Boreal,ikleyn)

Given that {{{ x^2-3x+2 }}} is a factor of {{{ x^4 + kx^3 - 10x^2 - 20x+24 }}} evaluate... (answered by ikleyn,MathTherapy)

To Factor a trinomial that is quadratic in form ?... (answered by palanisamy)

I need to find all the factors of the polynomial: 3x^3+20x^2+23x-10. I already know that... (answered by richwmiller)

Factor completely. 20x^2 � 3x � 9 (answered by MathTherapy)

Show that x-1, either is or is not a factor of P(x)=2x^4 + 3x^3 - 12x^2 - 7x +6.... (answered by robertb)

Factor each of the following polynomials 1.2x^2 - 8x 2.-3s^2 + 9s 3.4x + 20x^2 4.5t-... (answered by 428225,Cromlix,ikleyn)