Question
Use the long division method to find the result when 3, x, cubed, minus, 4, x, squared, minus, 22, x, minus, 203x
3
−4x
2
−22x−20 is divided by 3, x, plus, 23x+2. If there is a remainder, express the result in the form q, left parenthesis, x, right parenthesis, plus, start fraction, r, left parenthesis, x, right parenthesis, divided by, b, left parenthesis, x, right parenthesis, end fractionq(x)+
b(x)
r(x)
.
Solve the problem just like the example below
⬇️
Use the long division method to find the result when
4, x, cubed, minus, 20, x, squared, minus, 27, x, plus, 19
4x
3
−20x
2
−27x+19 is divided by
x, minus, 6
x−6. If there is a remainder, express the result in the form
q, left parenthesis, x, right parenthesis, plus, start fraction, r, left parenthesis, x, right parenthesis, divided by, b, left parenthesis, x, right parenthesis, end fraction
q(x)+
b(x)
r(x)
.
Use Long Division:
Use Long Division:
4, x, squared
4x
2
plus, 4, x
+4x
minus, 3
−3
x
x
minus, 6
−6
right parenthesis
)
4, x, cubed
4x
3
minus, 20, x, squared
−20x
2
minus, 27, x
−27x
plus, 19
+19
start superscript, minus, end superscript, 4, x, cubed
−
4x
3
minus, start superscript, plus, end superscript, 24, x, squared
−
+
24x
2
4, x, squared
4x
2
minus, 27, x
−27x
plus, 19
+19
start superscript, minus, end superscript, 4, x, squared
−
4x
2
minus, start superscript, plus, end superscript, 24, x
−
+
24x
minus, 3, x
−3x
plus, 19
+19
minus, start superscript, plus, end superscript, 3, x
−
+
3x
plus, start superscript, minus, end superscript, 18
+
−
18
1
1
Answer: , 4, x, squared, plus, 4, x, minus, 3, plus, start fraction, 1, divided by, x, minus, 6, end fraction
Answer: 4x
2
+4x−3+
x−6
1
Form: Quotient plus , start fraction, Remainder, divided by, Divisor, end fraction
Form: Quotient +
Divisor
Remainder
3
−4x
2
−22x−20 is divided by 3, x, plus, 23x+2. If there is a remainder, express the result in the form q, left parenthesis, x, right parenthesis, plus, start fraction, r, left parenthesis, x, right parenthesis, divided by, b, left parenthesis, x, right parenthesis, end fractionq(x)+
b(x)
r(x)
.
Solve the problem just like the example below
⬇️
Use the long division method to find the result when
4, x, cubed, minus, 20, x, squared, minus, 27, x, plus, 19
4x
3
−20x
2
−27x+19 is divided by
x, minus, 6
x−6. If there is a remainder, express the result in the form
q, left parenthesis, x, right parenthesis, plus, start fraction, r, left parenthesis, x, right parenthesis, divided by, b, left parenthesis, x, right parenthesis, end fraction
q(x)+
b(x)
r(x)
.
Use Long Division:
Use Long Division:
4, x, squared
4x
2
plus, 4, x
+4x
minus, 3
−3
x
x
minus, 6
−6
right parenthesis
)
4, x, cubed
4x
3
minus, 20, x, squared
−20x
2
minus, 27, x
−27x
plus, 19
+19
start superscript, minus, end superscript, 4, x, cubed
−
4x
3
minus, start superscript, plus, end superscript, 24, x, squared
−
+
24x
2
4, x, squared
4x
2
minus, 27, x
−27x
plus, 19
+19
start superscript, minus, end superscript, 4, x, squared
−
4x
2
minus, start superscript, plus, end superscript, 24, x
−
+
24x
minus, 3, x
−3x
plus, 19
+19
minus, start superscript, plus, end superscript, 3, x
−
+
3x
plus, start superscript, minus, end superscript, 18
+
−
18
1
1
Answer: , 4, x, squared, plus, 4, x, minus, 3, plus, start fraction, 1, divided by, x, minus, 6, end fraction
Answer: 4x
2
+4x−3+
x−6
1
Form: Quotient plus , start fraction, Remainder, divided by, Divisor, end fraction
Form: Quotient +
Divisor
Remainder
Answers
GPT-4o mini
To find the result when \(3x^3 - 4x^2 - 22x - 203\) is divided by \(3x + 2\) using long division, we follow these steps:
### Step 1: Setup the Long Division
```
_________________________
3x + 2 | 3x^3 - 4x^2 - 22x - 203
```
### Step 2: Perform the Long Division
1. **First Division**: Divide the leading term of the dividend \(3x^3\) by the leading term of the divisor \(3x\):
\[
\frac{3x^3}{3x} = x^2
\]
Write \(x^2\) above the division bar.
2. **Multiply and Subtract**: Multiply the entire divisor \(3x + 2\) by \(x^2\):
\[
x^2(3x + 2) = 3x^3 + 2x^2
\]
Subtract this from the original polynomial:
\[
(3x^3 - 4x^2 - 22x - 203) - (3x^3 + 2x^2) = -6x^2 - 22x - 203
\]
3. **Next Division**: Divide the new leading term \(-6x^2\) by \(3x\):
\[
\frac{-6x^2}{3x} = -2x
\]
Write \(-2x\) above the division bar.
4. **Multiply and Subtract**: Multiply the divisor by \(-2x\):
\[
-2x(3x + 2) = -6x^2 - 4x
\]
Subtract this from \(-6x^2 - 22x - 203\):
\[
(-6x^2 - 22x - 203) - (-6x^2 - 4x) = -18x - 203
\]
5. **Next Division**: Divide the new leading term \(-18x\) by \(3x\):
\[
\frac{-18x}{3x} = -6
\]
Write \(-6\) above the division bar.
6. **Multiply and Subtract**: Multiply the divisor by \(-6\):
\[
-6(3x + 2) = -18x - 12
\]
Subtract this from \(-18x - 203\):
\[
(-18x - 203) - (-18x - 12) = -191
\]
### Step 3: Write the Result
At this point, all the terms of the original polynomial have been accounted for, and we are left with:
- Quotient: \(x^2 - 2x - 6\)
- Remainder: \(-191\)
Putting it all together, the result of the division can be expressed as:
\[
x^2 - 2x - 6 + \frac{-191}{3x + 2}
\]
### Final Answer
The final result when \(3x^3 - 4x^2 - 22x - 203\) is divided by \(3x + 2\) is:
\[
\boxed{x^2 - 2x - 6 - \frac{191}{3x + 2}}
\]
### Step 1: Setup the Long Division
```
_________________________
3x + 2 | 3x^3 - 4x^2 - 22x - 203
```
### Step 2: Perform the Long Division
1. **First Division**: Divide the leading term of the dividend \(3x^3\) by the leading term of the divisor \(3x\):
\[
\frac{3x^3}{3x} = x^2
\]
Write \(x^2\) above the division bar.
2. **Multiply and Subtract**: Multiply the entire divisor \(3x + 2\) by \(x^2\):
\[
x^2(3x + 2) = 3x^3 + 2x^2
\]
Subtract this from the original polynomial:
\[
(3x^3 - 4x^2 - 22x - 203) - (3x^3 + 2x^2) = -6x^2 - 22x - 203
\]
3. **Next Division**: Divide the new leading term \(-6x^2\) by \(3x\):
\[
\frac{-6x^2}{3x} = -2x
\]
Write \(-2x\) above the division bar.
4. **Multiply and Subtract**: Multiply the divisor by \(-2x\):
\[
-2x(3x + 2) = -6x^2 - 4x
\]
Subtract this from \(-6x^2 - 22x - 203\):
\[
(-6x^2 - 22x - 203) - (-6x^2 - 4x) = -18x - 203
\]
5. **Next Division**: Divide the new leading term \(-18x\) by \(3x\):
\[
\frac{-18x}{3x} = -6
\]
Write \(-6\) above the division bar.
6. **Multiply and Subtract**: Multiply the divisor by \(-6\):
\[
-6(3x + 2) = -18x - 12
\]
Subtract this from \(-18x - 203\):
\[
(-18x - 203) - (-18x - 12) = -191
\]
### Step 3: Write the Result
At this point, all the terms of the original polynomial have been accounted for, and we are left with:
- Quotient: \(x^2 - 2x - 6\)
- Remainder: \(-191\)
Putting it all together, the result of the division can be expressed as:
\[
x^2 - 2x - 6 + \frac{-191}{3x + 2}
\]
### Final Answer
The final result when \(3x^3 - 4x^2 - 22x - 203\) is divided by \(3x + 2\) is:
\[
\boxed{x^2 - 2x - 6 - \frac{191}{3x + 2}}
\]