3.3 Long Division of Polynomials In Exercises 51-56, use long division to divide. 51. \( \frac{30 x^{2}-3 x+8}{5 x-3} \) 52. \( \frac{4 x+7}{3 x-2} \)

To perform long division of polynomials, we divide the numerator by the denominator. Let's start with the first division: \( \frac{30 x^{2}-3 x+8}{5 x-3} \) 1. Divide the leading term of the numerator by the leading term of the denominator: \( \frac{30x^2}{5x} = 6x \) 2. Multiply the entire denominator by the result from step 1: \( (5x-3) \times 6x = 30x^2 - 18x \) 3. Subtract the result from step 2 from the numerator: \( (30x^2 - 3x + 8) - (30x^2 - 18x) = 15x + 8 \) 4. Bring down the next term from the numerator: \( 15x + 8 \) 5. Repeat the process until the degree of the remainder is less than the degree of the denominator. Let's perform the long division for the second division: \( \frac{4x+7}{3x-2} \) 1. Divide the leading term of the numerator by the leading term of the denominator: \( \frac{4x}{3x} = \frac{4}{3} \) 2. Multiply the entire denominator by the result from step 1: \( (3x-2) \times \frac{4}{3} = 4x - \frac{8}{3} \) 3. Subtract the result from step 2 from the numerator: \( (4x+7) - (4x - \frac{8}{3}) = \frac{29}{3} \) 4. Bring down the next term from the numerator: \( \frac{29}{3} \) 5. Repeat the process until the degree of the remainder is less than the degree of the denominator. Let's calculate the results for each division. Divide the polynomials by following steps: - step0: Using Long Division to Divide Polynomials: \(\frac{\left(30x^{2}-3x+8\right)}{\left(5x-3\right)}\) - step1: Calculate: \(6x+3+\frac{17}{5x-3}\) Calculate or simplify the expression \( (4x+7)/(3x-2) \). Divide the polynomials by following steps: - step0: Using Long Division to Divide Polynomials: \(\frac{\left(4x+7\right)}{\left(3x-2\right)}\) - step1: Calculate: \(\frac{4}{3}+\frac{\frac{29}{3}}{3x-2}\) The results of the long division of polynomials are: 1. \( \frac{30 x^{2}-3 x+8}{5 x-3} = 6x + 3 + \frac{17}{5x-3} \) 2. \( \frac{4 x+7}{3 x-2} = \frac{4}{3} + \frac{\frac{29}{3}}{3x-2} \)