Question: For what value of 'a' does the polynomial 2x-1 divide the polynomial 4x^3 + ax + 1? Justify.
If (2x - 1) divides the polynomial 4x^3 + ax + 1, then that polynomial is factorisable, and can be factored into the form shown below, the product of a linear term and a quadratic term.
(px - 1)(qx^2 + rx - 1) = pqx^3 + prx ^2 - px - qx^2 - rx + 1 = pqx^3 +(pr - q)x^2 - (p+r)x - 1
(px - 1)(qx^2 + rx - 1) = pqx^3 +(pr - q)x^2 - (p+r)x - 1
where p = 2, pq = 4, hence q = 2, giving
(2x - 1)(2x^2 + rx - 1) = 4x^3 + (2r - 2)x^2 - (2+r)x - 1
in order to exclude the x^2-term, we need r = 1, this gives the final expression as,
(2x - 1)(2x^2 + x - 1) = 4x^3 - (3)x + 1
Answer: a = -3