( 3. )(20 c^3+22 c^2-7 c+7) div(5 c-2) 5 c-2 sqrt(20 c^3)+22 c^(2-7 c+)

To divide the polynomial \(20c^3 + 22c^2 - 7c + 7\) by \(5c - 2\), we can use polynomial long division.<br /><br />Step 1: Divide the leading term of the dividend by the leading term of the divisor.<br />\(\frac{20c^3}{5c} = 4c^2\)<br /><br />Step 2: Multiply the divisor by the result from Step 1 and subtract it from the dividend.<br />\((5c - 2) \cdot 4c^2 = 20c^3 - 8c^2\)<br />\(20c^3 + 22c^2 - 7c + 7 - (20c^3 - 8c^2) = 30c^2 - 7c + 7\)<br /><br />Step 3: Repeat the process with the new dividend.<br />\(\frac{30c^2}{5c} = 6c\)<br /><br />Step 4: Multiply the divisor by the result from Step 3 and subtract it from the new dividend.<br />\((5c - 2) \cdot 6c = 30c^2 - 12c\)<br />\(30c^2 - 7c + 7 - (30c^2 - 12c) = 5c + 7\)<br /><br />Step 5: Repeat the process with the new dividend.<br />\(\frac{5c}{5c} = 1\)<br /><br />Step 6: Multiply the divisor by the result from Step 5 and subtract it from the new dividend.<br />\((5c - 2) \cdot 1 = 5c - 2\)<br />\(5c + 7 - (5c - 2) = 9\)<br /><br />So, the quotient is \(4c^2 + 6c + 1\) and the remainder is \(9\).<br /><br />Therefore, the answer is:<br />\(4c^2 + 6c + 1 + \frac{9}{5c - 2}\)