Question

4x
3
+4x
2
+19x+9 is divided by 2, x, plus, 12x+1.

Answer 1

I want to be sure I understand — do you mean the polynomial
P(x) = 4x^3 + 4x^2 + 19x + 9
is to be divided by 2x + 1 and by 12x + 1 (or by their product)?

If so, here are the results:

- Remainder on division by 2x + 1:
Evaluate P at x = −1/2 → P(−1/2) = 0. So 2x + 1 is a factor; the remainder is 0.
The quotient is 2x^2 + x + 9 (since P(x) = (2x+1)(2x^2 + x + 9)).

- Remainder on division by 12x + 1:
Evaluate P at x = −1/12 → P(−1/12) = 3215/432. So the remainder is 3215/432.

- If dividing by the product (2x+1)(12x+1) = 24x^2 + 14x + 1:
The remainder must be linear: R(x) = Ax + B. Matching values at x = −1/2 and x = −1/12 gives
R(x) = (643/36)x + 643/72, which can be written R(x) = (643/72)(2x+1).

Tell me which division you want worked out step-by-step (long division, synthetic, or quotient by the product) and I’ll show it.