The discussion revolves around the methods for factoring the polynomial $6x^4+17x^3-24x^2-53x+30$ without employing the rational root theorem. Participants explore various techniques and approaches to achieve this goal.
Participants generally agree on the factorization method presented, but there is no consensus on the specific techniques used to identify the necessary pairs of numbers for rewriting the polynomial.
The discussion does not clarify the underlying assumptions or limitations of the methods used, nor does it address the potential for alternative approaches to factoring the polynomial.
MarkFL said:You could write:
$$6x^4+17x^3-24x^2-53x+30=\left(6x^4+12x^3\right)+\left(5x^3+10x^2\right)-\left(34x^2+68x\right)+\left(15x+30\right)=6x^3(x+2)+5x^2(x+2)-34x(x+2)+15(x+2)=(x+2)\left(6x^3+5x^2-34x+15\right)$$
$$6x^3+5x^2-34x+15=\left(6x^3+18x^2\right)-\left(13x^2+39x\right)+\left(5x+15\right)=6x^2(x+3)-13x(x+3)+5(x+3)=(x+3)\left(6x^2-13x+5\right)$$
$$6x^2-13x+5=(2x-1)(3x-5)$$
And so we have:
$$6x^4+17x^3-24x^2-53x+30=(x+2)(x+3)(2x-1)(3x-5)$$
paulmdrdo said:What technique did you use to determine the pair of numbers to be used to rewrite the expression? Was it by trial and error?