Questions: 6. Divide. Use long division. a) (6 x^2 + 10 x - 5) ÷ (3 x - 1) e) (-3 x + 2 x^3 + 3 x^2 + 4) ÷ (x + 2)

Solution Steps

To solve these polynomial division problems using long division, we will follow these steps:

1. Arrange the terms of both the dividend and the divisor in descending order of their degrees.
2. Divide the leading term of the dividend by the leading term of the divisor to get the first term of the quotient.
3. Multiply the entire divisor by this term and subtract the result from the dividend.
4. Repeat the process with the new polynomial obtained after subtraction until the degree of the remainder is less than the degree of the divisor.

We need to divide \( 6x^2 + 10x - 5 \) by \( 3x - 1 \).

1. Divide the leading term: \[ \frac{6x^2}{3x} = 2x \]
2. Multiply the entire divisor by \( 2x \): \[ 2x(3x - 1) = 6x^2 - 2x \]
3. Subtract this from the original polynomial: \[ (6x^2 + 10x - 5) - (6x^2 - 2x) = 12x - 5 \]
4. Now divide \( 12x - 5 \) by \( 3x - 1 \): \[ \frac{12x}{3x} = 4 \]
5. Multiply the divisor by \( 4 \): \[ 4(3x - 1) = 12x - 4 \]
6. Subtract again: \[ (12x - 5) - (12x - 4) = -1 \]

Thus, the result for part a is: \[ \text{Quotient: } 2x + 4, \quad \text{Remainder: } -1 \]

Now we divide \( 2x^3 + 3x^2 - 3x + 4 \) by \( x + 2 \).

1. Divide the leading term: \[ \frac{2x^3}{x} = 2x^2 \]
2. Multiply the entire divisor by \( 2x^2 \): \[ 2x^2(x + 2) = 2x^3 + 4x^2 \]
3. Subtract this from the original polynomial: \[ (2x^3 + 3x^2 - 3x + 4) - (2x^3 + 4x^2) = -x^2 - 3x + 4 \]
4. Now divide \( -x^2 - 3x + 4 \) by \( x + 2 \): \[ \frac{-x^2}{x} = -x \]
5. Multiply the divisor by \( -x \): \[ -x(x + 2) = -x^2 - 2x \]
6. Subtract again: \[ (-x^2 - 3x + 4) - (-x^2 - 2x) = -x + 4 \]
7. Finally, divide \( -x + 4 \) by \( x + 2 \): \[ \frac{-x}{x} = -1 \]
8. Multiply the divisor by \( -1 \): \[ -1(x + 2) = -x - 2 \]
9. Subtract: \[ (-x + 4) - (-x - 2) = 6 \]

Thus, the result for part e is: \[ \text{Quotient: } 2x^2 - x - 1, \quad \text{Remainder: } 6 \]

Final Answer