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. $$ \frac{4 x+7}{3 x-2} $$

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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:
$$ (5x-3) 	imes 6x = 30x^2 - 18x $$

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:
$$ (3x-2) 	imes \frac{4}{3} = 4x - \frac{8}{3} $$

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:
  $$\frac{\left(30x^{2}-3x+8ight)}{\left(5x-3ight)}$$
- 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:
  $$\frac{\left(4x+7ight)}{\left(3x-2ight)}$$
- 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. $$ \frac{4 x+7}{3 x-2} = \frac{4}{3} + \frac{\frac{29}{3}}{3x-2} $$
The results of the long divisions are: 1. $$ \frac{30x^{2}-3x+8}{5x-3} = 6x + 3 + \frac{17}{5x-3} $$ 2. $$ \frac{4x+7}{3x-2} = \frac{4}{3} + \frac{29}{3(3x-2)} $$

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. \( \frac{4 x+7}{3 x-2} \)

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