Given that p(x) leaves a remainder of R when divided by (x-1), we have p(1) = R.

Hey Ria, good to see you again. a) Find the values of R and k. Step 1: Use the Remainder Theorem for division by (x-1). Given that p(x) leaves a remainder of R when divided by (x-1), we have p(1) = R. Substitute x=1 into p(x) = x^3 + kx^2 - x + 12: p(1) = (1)^3 + k(1)^2 - (1) + 12 R = 1 + k - 1 + 12 R = k + 12 (*) Step 2: Use the Remainder Theorem for division by (x-4). Given that p(x) leaves a remainder of 8R when divided by (x-4), we have p(4) = 8R. Substitute x=4 into p(x) = x^3 + kx^2 - x + 12: p(4) = (4)^3 + k(4)^2 - (4) + 12 8R = 64 + 16k - 4 + 12 8R = 16k + 72 (**) Step 3: Solve the system of equations for R and k. Substitute equation () into equation (*): 8(k + 12) = 16k + 72 8k + 96 = 16k + 72 96 - 72 = 16k - 8k 24 = 8k k = (24)/(8) k = 3 Now substitute k=3 back into equation (*) to find R: R = 3 + 12 R = 15 The values are R=15, k=3. b) Show that (x+4) is a factor of p(x). Step 1: Update p(x) with the value of k. Using k=3, the polynomial is p(x) = x^3 + 3x^2 - x + 12. Step 2: Apply the Factor Theorem. For (x+4) to be a factor of p(x), we must show that p(-4) = 0. Substitute x=-4 into p(x): p(-4) = (-4)^3 + 3(-4)^2 - (-4) + 12 p(-4) = -64 + 3(16) + 4 + 12 p(-4) = -64 + 48 + 4 + 12 p(-4) = -64 + 64 p(-4) = 0 Since p(-4) = 0, by the Factor Theorem, (x+4) is a factor of p(x). Send me the next one 📸