Use the long division method to find the result when 4, x, cubed, minus, x, squared, minus, 13, x, plus, 104x

Question

3
−x
2
−13x+10 is divided by x, minus, 1x−1.

Use the long division method to find the result when
4, x, cubed, plus, 12, x, squared, plus, 27, x, plus, 27
4x
3
+12x
2
+27x+27 is divided by
2, x, plus, 3
2x+3.
Use Long Division:
Use Long Division:

2, x, squared
2x
2
plus, 3, x
+3x
plus, 9
+9

2, x
2x
plus, 3
+3
right parenthesis
)
4, x, cubed
4x
3
plus, 12, x, squared
+12x
2
plus, 27, x
+27x
plus, 27
+27

start superscript, minus, end superscript, 4, x, cubed
−
4x
3
plus, start superscript, minus, end superscript, 6, x, squared
+
−
6x
2

6, x, squared
6x
2
plus, 27, x
+27x
plus, 27
+27

start superscript, minus, end superscript, 6, x, squared
−
6x
2
plus, start superscript, minus, end superscript, 9, x
+
−
9x

18, x
18x
plus, 27
+27

start superscript, minus, end superscript, 18, x
−
18x
plus, start superscript, minus, end superscript, 27
+
−
27

0
0

Answer: , 2, x, squared, plus, 3, x, plus, 9
Answer: 2x
2
+3x+9

Answer 1

Let's perform long division for both cases you've provided.

First Case: Dividing \( 4x^3 - x^2 - 13x + 104 \) by \( x - 1 \)

1. Divide the leading term: \( \frac{4x^3}{x} = 4x^2 \).
2. Multiply: \( 4x^2(x - 1) = 4x^3 - 4x^2 \).
3. Subtract: \[ (4x^3 - x^2) - (4x^3 - 4x^2) = 3x^2 \]
4. Bring down the next term: Get \( 3x^2 - 13x \).
5. Repeat the process:
   • Divide \( \frac{3x^2}{x} = 3x \).
   • Multiply: \( 3x(x - 1) = 3x^2 - 3x \).
   • Subtract: \[ (3x^2 - 13x) - (3x^2 - 3x) = -10x \]
6. Bring down the next term: Get \( -10x + 104 \).
7. Repeat the process:
   • Divide \( \frac{-10x}{x} = -10 \).
   • Multiply: \( -10(x - 1) = -10x + 10 \).
   • Subtract: \[ (-10x + 104) - (-10x + 10) = 94 \]

The result of the division is: \[ 4x^2 + 3x - 10 \quad \text{with a remainder of } 94. \]

Second Case: Dividing \( 4x^3 + 12x^2 + 27x + 27 \) by \( 2x + 3 \)

1. Divide the leading term: \( \frac{4x^3}{2x} = 2x^2 \).
2. Multiply: \( 2x^2(2x + 3) = 4x^3 + 6x^2 \).
3. Subtract: \[ (4x^3 + 12x^2) - (4x^3 + 6x^2) = 6x^2 \]
4. Bring down the next term: Get \( 6x^2 + 27x \).
5. Repeat the process:
   • Divide \( \frac{6x^2}{2x} = 3x \).
   • Multiply: \( 3x(2x + 3) = 6x^2 + 9x \).
   • Subtract: \[ (6x^2 + 27x) - (6x^2 + 9x) = 18x \]
6. Bring down the next term: Get \( 18x + 27 \).
7. Repeat the process:
   • Divide \( \frac{18x}{2x} = 9 \).
   • Multiply: \( 9(2x + 3) = 18x + 27 \).
   • Subtract: \[ (18x + 27) - (18x + 27) = 0 \]

The result of the division is: \[ 2x^2 + 3x + 9 \quad \text{with a remainder of } 0. \]

Final Answers:

1. For the first division: \( 4x^2 + 3x - 10 \) with a remainder of \( 94 \).
2. For the second division: \( 2x^2 + 3x + 9 \) with a remainder of \( 0 \).