Question

Use the long division method to find the result when 9, x, cubed, plus, 12, x, squared, plus, 25, x, plus, 149x
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:
\[
\text{Quotient: } 3x^2 + 2x + 7, \quad \text{Remainder: } 135
\]