Use the long division method to find the result when 9, x, cubed, plus, 12, x, squared, plus, 25, x, plus, 149x

Question

3
+12x
2
+25x+14 is divided by 3, x, plus, 23x+2.

Solve the problem just like the example below
⬇️
Use the long division method to find the result when
4, x, cubed, plus, 5, x, squared, plus, 5, x, plus, 1
4x
3
+5x
2
+5x+1 is divided by
4, x, plus, 1
4x+1.
Use Long Division:
Use Long Division:

x, squared
x
2
plus, x
+x
plus, 1
+1

4, x
4x
plus, 1
+1
right parenthesis
)
4, x, cubed
4x
3
plus, 5, x, squared
+5x
2
plus, 5, x
+5x
plus, 1
+1

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

4, x, squared
4x
2
plus, 5, x
+5x
plus, 1
+1

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

4, x
4x
plus, 1
+1

start superscript, minus, end superscript, 4, x
−
4x
plus, start superscript, minus, end superscript, 1
+
−
1

0
0

Answer: , x, squared, plus, x, plus, 1
Answer: x
2
+x+1

Answer 1

Let's divide \( 9x^3 + 12x^2 + 25x + 149 \) by \( 3x + 2 \) using the long division method.

Step 1: Set up the long division.

                _____________
3x + 2     |   9x^3 + 12x^2 + 25x + 149 

Step 2: Divide the leading term.

1. Divide the leading term of the dividend \( 9x^3 \) by the leading term of the divisor \( 3x \). \[ \frac{9x^3}{3x} = 3x^2 \]

2. Write \( 3x^2 \) above the division line.

                3x^2         
                _____________
3x + 2     |   9x^3 + 12x^2 + 25x + 149 

Step 3: Multiply and subtract.

1. Multiply \( 3x^2 \) by the entire divisor \( 3x + 2 \). \[ 3x^2 \cdot (3x + 2) = 9x^3 + 6x^2 \]

2. Write this under the dividend and subtract.

                3x^2         
                _____________
3x + 2     |   9x^3 + 12x^2 + 25x + 149 
                -(9x^3 + 6x^2)
                __________________
                         6x^2 + 25x + 149       

Step 4: Bring down and repeat.

1. Bring down the next term to get \( 6x^2 + 25x \) under the line.

2. Divide \( 6x^2 \) by \( 3x \). \[ \frac{6x^2}{3x} = 2x \]

3. Write \( 2x \) above the division line.

                3x^2 + 2x        
                ________________
3x + 2     |   9x^3 + 12x^2 + 25x + 149 
                -(9x^3 + 6x^2)
                __________________
                        6x^2 + 25x + 149          
                       -(6x^2 + 4x)
                __________________
                             21x + 149       

Step 5: Multiply and subtract again.

1. Multiply \( 2x \) by the entire divisor \( 3x + 2 \). \[ 2x \cdot (3x + 2) = 6x^2 + 4x \]

2. Subtract this from the current remainder.

                3x^2 + 2x        
                ________________
3x + 2     |   9x^3 + 12x^2 + 25x + 149 
                -(9x^3 + 6x^2)
                __________________
                        6x^2 + 25x + 149          
                       -(6x^2 + 4x)
                __________________
                             21x + 149       

Step 6: Divide again.

1. Divide \( 21x \) by \( 3x \). \[ \frac{21x}{3x} = 7 \]

2. Write \( 7 \) above the line.

                3x^2 + 2x + 7 
                ________________
3x + 2     |   9x^3 + 12x^2 + 25x + 149 
                -(9x^3 + 6x^2)
                __________________
                        6x^2 + 25x + 149          
                       -(6x^2 + 4x)
                __________________
                             21x + 149       
                             -(21x + 14)
                __________________
                                  135

Step 7: Final Remainder

1. Multiply \( 7 \) by the divisor \( 3x + 2 \). \[ 7 \cdot (3x + 2) = 21x + 14 \]

2. Subtract this from \( 21x + 149 \).

Resulting in 135.

Final Result

The quotient is: \[ 3x^2 + 2x + 7 \] And the remainder is: \[ 135 \]

So the answer for when dividing \( 9x^3 + 12x^2 + 25x + 149 \) by \( 3x + 2 \) is: \[ \text{Quotient: } 3x^2 + 2x + 7, \quad \text{Remainder: } 135 \]