Divide using long division

Question

1. x⁴ - x² - 7/ x + 4

2. 12x⁴ - 25x³ + 6x² + 11x - 4 ÷ 3x² - 7x + 4

3. 2x³ + 5x² + 17x + 21 ÷ 2x + 3

4. x³ - 10x² + 21x - 1/ x-2

5. x³ + 7x² + 15x + 9 ÷ x + 1

6. x⁴+ 4x³ - x - 4 ÷ x³ - 1

7. 3x⁴ - 13x³ + 29x² - 16x - 28/ x² - 3x + 7

Answer 1

Here are the results of the long division for each of the given polynomial expressions:

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

1. Divide \( x^4 \) by \( x \) to get \( x^3 \).
2. Multiply \( x^3 \) by \( x + 4\) to get \( x^4 + 4x^3 \).
3. Subtract: \( (x^4 - x^2 - 7) - (x^4 + 4x^3) = -4x^3 - x^2 - 7 \).
4. Divide \( -4x^3 \) by \( x \) to get \( -4x^2 \).
5. Multiply: \( -4x^2(x + 4) = -4x^3 - 16x^2 \).
6. Subtract: \( (-4x^3 - x^2 - 7) - (-4x^3 - 16x^2) = 15x^2 - 7 \).
7. Divide \( 15x^2 \) by \( x \) to get \( 15x \).
8. Multiply: \( 15x(x + 4) = 15x^2 + 60x \).
9. Subtract: \( (15x^2 - 7) - (15x^2 + 60x) = -60x - 7 \).
10. Divide \( -60x \) by \( x \) to get \( -60 \).
11. Multiply: \( -60(x + 4) = -60x - 240 \).
12. Subtract: \( (-60x - 7) - (-60x - 240) = 233 \).

Final result: \[ \frac{x^4 - x^2 - 7}{x + 4} = x^3 - 4x^2 + 15x - 60 + \frac{233}{x + 4}. \]

2. \( \frac{12x^4 - 25x^3 + 6x^2 + 11x - 4}{3x^2 - 7x + 4} \)

1. Divide \( 12x^4 \) by \( 3x^2 \) to get \( 4x^2 \).
2. Multiply: \( 4x^2(3x^2 - 7x + 4) = 12x^4 - 28x^3 + 16x^2 \).
3. Subtract: \( (12x^4 - 25x^3 + 6x^2 + 11x - 4) - (12x^4 - 28x^3 + 16x^2) = 3x^3 - 10x^2 + 11x - 4 \).
4. Divide \( 3x^3 \) by \( 3x^2 \) to get \( x \).
5. Multiply: \( x(3x^2 - 7x + 4) = 3x^3 - 7x^2 + 4x \).
6. Subtract: \( (3x^3 - 10x^2 + 11x - 4) - (3x^3 - 7x^2 + 4x) = -3x^2 + 7x - 4 \).
7. Divide \( -3x^2 \) by \( 3x^2 \) to get \( -1 \).
8. Multiply: \( -1(3x^2 - 7x + 4) = -3x^2 + 7x - 4 \).
9. Subtract: \( (-3x^2 + 7x - 4) - (-3x^2 + 7x - 4) = 0 \).

Final result: \[ \frac{12x^4 - 25x^3 + 6x^2 + 11x - 4}{3x^2 - 7x + 4} = 4x^2 + x - 1. \]

3. \( \frac{2x^3 + 5x^2 + 17x + 21}{2x + 3} \)

1. Divide \( 2x^3 \) by \( 2x \) to get \( x^2 \).
2. Multiply: \( x^2(2x + 3) = 2x^3 + 3x^2 \).
3. Subtract: \( (2x^3 + 5x^2 + 17x + 21) - (2x^3 + 3x^2) = 2x^2 + 17x + 21 \).
4. Divide \( 2x^2 \) by \( 2x \) to get \( x \).
5. Multiply: \( x(2x + 3) = 2x^2 + 3x \).
6. Subtract: \( (2x^2 + 17x + 21) - (2x^2 + 3x) = 14x + 21 \).
7. Divide \( 14x \) by \( 2x \) to get \( 7 \).
8. Multiply: \( 7(2x + 3) = 14x + 21 \).
9. Subtract: \( (14x + 21) - (14x + 21) = 0 \).

Final result: \[ \frac{2x^3 + 5x^2 + 17x + 21}{2x + 3} = x^2 + x + 7. \]

4. \( \frac{x^3 - 10x^2 + 21x - 1}{x - 2} \)

1. Divide \( x^3 \) by \( x \) to get \( x^2 \).
2. Multiply: \( x^2(x - 2) = x^3 - 2x^2 \).
3. Subtract: \( (x^3 - 10x^2 + 21x - 1) - (x^3 - 2x^2) = -8x^2 + 21x - 1 \).
4. Divide \( -8x^2 \) by \( x \) to get \( -8x \).
5. Multiply: \( -8x(x - 2) = -8x^2 + 16x \).
6. Subtract: \( (-8x^2 + 21x - 1) - (-8x^2 + 16x) = 5x - 1 \).
7. Divide \( 5x \) by \( x \) to get \( 5 \).
8. Multiply: \( 5(x - 2) = 5x - 10 \).
9. Subtract: \( (5x - 1) - (5x - 10) = 9 \).

Final result: \[ \frac{x^3 - 10x^2 + 21x - 1}{x - 2} = x^2 - 8x + 5 + \frac{9}{x - 2}. \]

5. \( \frac{x^3 + 7x^2 + 15x + 9}{x + 1} \)

1. Divide \( x^3 \) by \( x \) to get \( x^2 \).
2. Multiply: \( x^2(x + 1) = x^3 + x^2 \).
3. Subtract: \( (x^3 + 7x^2 + 15x + 9) - (x^3 + x^2) = 6x^2 + 15x + 9 \).
4. Divide \( 6x^2 \) by \( x \) to get \( 6x \).
5. Multiply: \( 6x(x + 1) = 6x^2 + 6x \).
6. Subtract: \( (6x^2 + 15x + 9) - (6x^2 + 6x) = 9x + 9 \).
7. Divide \( 9x \) by \( x \) to get \( 9 \).
8. Multiply: \( 9(x + 1) = 9x + 9 \).
9. Subtract: \( (9x + 9) - (9x + 9) = 0 \).

Final result: \[ \frac{x^3 + 7x^2 + 15x + 9}{x + 1} = x^2 + 6x + 9. \]

6. \( \frac{x^4 + 4x^3 - x - 4}{x^3 - 1} \)

1. Divide \( x^4 \) by \( x^3 \) to get \( x \).
2. Multiply: \( x(x^3 - 1) = x^4 - x \).
3. Subtract: \( (x^4 + 4x^3 - x - 4) - (x^4 - x) = 4x^3 - 4 \).
4. Divide \( 4x^3 \) by \( x^3 \) to get \( 4 \).
5. Multiply: \( 4(x^3 - 1) = 4x^3 - 4 \).
6. Subtract: \( (4x^3 - 4) - (4x^3 - 4) = 0 \).

Final result: \[ \frac{x^4 + 4x^3 - x - 4}{x^3 - 1} = x + 4. \]

7. \( \frac{3x^4 - 13x^3 + 29x^2 - 16x - 28}{x^2 - 3x + 7} \)

1. Divide \( 3x^4 \) by \( x^2 \) to get \( 3x^2 \).
2. Multiply: \( 3x^2(x^2 - 3x + 7) = 3x^4 - 9x^3 + 21x^2 \).
3. Subtract: \( (3x^4 - 13x^3 + 29x^2 - 16x - 28) - (3x^4 - 9x^3 + 21x^2) = -4x^3 + 8x^2 - 16x - 28 \).
4. Divide \( -4x^3 \) by \( x^2 \) to get \( -4x \).
5. Multiply: \( -4x(x^2 - 3x + 7) = -4x^3 + 12x^2 - 28x \).
6. Subtract: \( (-4x^3 + 8x^2 - 16x - 28) - (-4x^3 + 12x^2 - 28x) = -4x^2 + 12x - 28 \).
7. Divide \( -4x^2 \) by \( x^2 \) to get \( -4 \).
8. Multiply: \( -4(x^2 - 3x + 7) = -4x^2 + 12x - 28 \).
9. Subtract: \( (-4x^2 + 12x - 28) - (-4x^2 + 12x - 28) = 0 \).

Final result: \[ \frac{3x^4 - 13x^3 + 29x^2 - 16x - 28}{x^2 - 3x + 7} = 3x^2 - 4x - 4. \]

If you need further elaboration or any specific exploration of one of the problems, feel free to ask!