Given that when P(x) is divided by (x+2), the remainder is 3, we have P(-2) = 3.

Hey ThegirlGodishelping🥰💙🫠, good to see you again. Step 1: Apply the Remainder Theorem. Let the polynomial be P(x). According to the Remainder Theorem, if a polynomial P(x) is divided by (x-c), the remainder is P(c). Given that when P(x) is divided by (x+2), the remainder is 3, we have P(-2) = 3. Given that when P(x) is divided by (x-1), the remainder is -1, we have P(1) = -1. Step 2: Express the polynomial in terms of the new divisor and remainder. When P(x) is divided by (x+2)(x-1), the divisor is a quadratic polynomial. Therefore, the remainder must be a linear polynomial of the form Ax + B. We can write P(x) as: P(x) = Q(x)(x+2)(x-1) + (Ax + B) where Q(x) is the quotient and Ax + B is the remainder. Step 3: Use the given conditions to form a system of equations. Substitute x = -2 into the equation for P(x): P(-2) = Q(-2)(-2+2)(-2-1) + (A(-2) + B) 3 = Q(-2)(0)(-3) + (-2A + B) 3 = -2A + B (*) Substitute x = 1 into the equation for P(x): P(1) = Q(1)(1+2)(1-1) + (A(1) + B) -1 = Q(1)(3)(0) + (A + B) -1 = A + B (**) Step 4: Solve the system of linear equations for A and B. We have the system: 1) -2A + B = 3 2) A + B = -1 Subtract equation (2) from equation (1): (-2A + B) - (A + B) = 3 - (-1) -2A - A + B - B = 3 + 1 -3A = 4 A = -(4)/(3) Substitute the value of A into equation (2): -(4)/(3) + B = -1 B = -1 + (4)/(3) B = -(3)/(3) + (4)/(3) B = (1)/(3) Step 5: State the remainder. The remainder is Ax + B. Substitute the values of A and B: Remainder = -(4)/(3)x + (1)/(3) The remainder when the polynomial is divided by (x+2)(x-1) is -(4)/(3)x + (1)/(3). Send me the next one 📸