Final answer:
To find the remainder of the division of the polynomial (3x^4 + 2x^3 - x^2 + 2x - 9) by the binomial (x + 2), we evaluate the polynomial at x = -2, which results in a remainder of 15.
Step-by-step explanation:
The question asks about the remainder when the polynomial (3x^4 + 2x^3 - x^2 + 2x - 9) is divided by the binomial (x + 2). To find this, we can use polynomial long division or synthetic division. The remainder theorem tells us that the remainder of this division is the value of the polynomial evaluated at the negation of the constant term of the divisor, meaning we would evaluate the polynomial at x = -2.
Substituting x = -2 into the polynomial gives us:
3(-2)^4 + 2(-2)^3 - (-2)^2 + 2(-2) - 9
= 3(16) - 2(8) - 4 - 4 - 9
= 48 - 16 - 4 - 4 - 9
= 48 - 33
= 15. So, the remainder is 15.