Check Your Understanding - Long Division Rafael Mestrener DIRECTIONS: Use this information to answer Parts A and B. Consider the problem: (3x^3-x^2+5)div (x^2-4x+1) Part A Find the quotient and remainder of (3x^3-x^2+5)div (x^2-4x+1) Enter the correct value in the box. Give your answer in the form dividend =(divisor)(quotient)+remainder. square

To find the quotient and remainder of $(3x^{3}-x^{2}+5)\div (x^{2}-4x+1)$, we can use long division.<br /><br />Step 1: Divide the leading term of the dividend by the leading term of the divisor.<br />$\frac{3x^{3}}{x^{2}} = 3x$<br /><br />Step 2: Multiply the divisor by the quotient obtained in step 1.<br />$(x^{2}-4x+1)(3x) = 3x^{3}-12x^{2}+3x$<br /><br />Step 3: Subtract the result from step 2 from the dividend.<br />$(3x^{3}-x^{2}+5) - (3x^{3}-12x^{2}+3x) = 11x^{2}-3x+5$<br /><br />Step 4: Repeat steps 1-3 with the new dividend obtained in step 3.<br />Divide the leading term of the new dividend by the leading term of the divisor.<br />$\frac{11x^{2}}{x^{2}} = 11$<br /><br />Multiply the divisor by the quotient obtained in step 1.<br />$(x^{2}-4x+1)(11) = 11x^{2}-44x+11$<br /><br />Subtract the result from step 2 from the new dividend.<br />$(11x^{2}-3x+5) - (11x^{2}-44x+11) = 41x-6$<br /><br />Step 5: Since the degree of the new dividend (41x-6) is less than the degree of the divisor (x^{2}-4x+1), we stop here.<br /><br />Therefore, the quotient is 3x + 11 and the remainder is 41x - 6.<br /><br />So, the answer is:<br />$(3x^{3}-x^{2}+5) = (x^{2}-4x+1)(3x+11) + (41x-6)$