Zadanie 9, Paper 1, maj 2022, TZ2, Matematyka AA IB HL

Assume an integer k is a root. Rearranging gives 3(k³+2k²+3k)=−2. Since k³+2k²+3k is an integer, the left side is divisible by 3. The right side is not divisible by 3, a contradiction. Thus no integer root exists. Alternatively, under the same integer-root assumption, define F(x)=3x³+6x²+9x+2. Its derivative is 9(x+2/3)²+5>0. Since F(−1)=−4 and F(0)=2, its unique real zero lies strictly between −1 and 0, contradicting the integer-root assumption.

M1:explicit assumption of an integer root. M1:rearrange toward divisibility. A1:correct factorization/equation. R1:left side is an integer multiple of 3. R1:contradiction with −2, rejecting the assumption. Alternative calculus route: M1:assume an integer root. M1:attempt derivative to establish monotonicity. A1:correct positive derivative. R1:opposite signs F(−1)=−4,F(0)=2. R1:the unique zero is strictly between consecutive integers, a contradiction. Later marks remain independently available when the initial assumption mark is lost.