Question

What is the result when the polynomial 4r^3 - 19r^2 + 191r + 13 is divided by 4r + 1? If there is a remainder, express the result in the form q(2) + b(2).

A) Quotient: 4r^2 - 3r + 194; Remainder: 10
B) Quotient: 4r^2 - 3r + 190; Remainder: 14
C) Quotient: 4r^2 - 4r + 192; Remainder: 11
D) Quotient: 4r^2 - 2r + 196; Remainder: 12

Answer 1

To divide the polynomial 4r³ - 19r² + 191r + 13 by 4r + 1, use synthetic division. The quotient is 4r² - 3r + 214 and the remainder is -201 = 4(-2) + (-193). Hence the correct answer is option A

Step-by-step explanation:

To divide the polynomial 4r³ - 19r² + 191r + 13 by 4r + 1, we use synthetic division:

Step 1: Write the coefficients of the polynomial in descending order: 4, -19, 191, 13

Step 2: Write the opposite of the constant term of the divisor: -1

Step 3: Bring down the first coefficient: 4

Step 4: Multiply the constant term of the divisor (-1) by the coefficient brought down (4): -4

Step 5: Add the result from step 4 to the next coefficient: -19 + (-4) = -23

Step 6: Multiply the constant term of the divisor (-1) by the result from step 5 (-23): 23

Step 7: Add the result from step 6 to the next coefficient: 191 + 23 = 214

Step 8: Multiply the constant term of the divisor (-1) by the result from step 7 (214): -214

Step 9: Add the result from step 8 to the next coefficient: 13 + (-214) = -201

The quotient is 4r² - 3r + 214 and the remainder is -201. Expressing the remainder in the requested form, we have -201 = 4(-2) + (-193), so the correct answer is A) Quotient: 4r² - 3r + 214; Remainder: -201 = 4(-2) + (-193).