Question
Use the long division method to find the result when 4, x, cubed, plus, 25, x, squared, minus, 13, x, minus, 14x
3
+25x
2
−13x−1 is divided by 4, x, minus, 34x−3. 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
+25x
2
−13x−1 is divided by 4, x, minus, 34x−3. 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
Let's use the long division method to divide \( 4x^3 + 25x^2 - 13x - 14 \) by \( 4x - 3 \).
### Step 1: Set Up the Long Division
We will divide \( 4x^3 + 25x^2 - 13x - 14 \) by \( 4x - 3 \).
```
_______________
4x - 3 | 4x^3 + 25x^2 - 13x - 14
```
### Step 2: Begin the Long Division
1. Divide the leading term of the dividend \( 4x^3 \) by the leading term of the divisor \( 4x \):
\[
\frac{4x^3}{4x} = x^2
\]
2. Multiply \( x^2 \) by \( 4x - 3 \) and subtract:
\[
x^2 (4x - 3) = 4x^3 - 3x^2
\]
```
x^2
_______________
4x - 3 | 4x^3 + 25x^2 - 13x - 14
-(4x^3 - 3x^2)
_______________
28x^2 - 13x - 14
```
### Step 3: Continue Dividing
3. Now bring down the \( -13x \), making it \( 28x^2 - 13x \).
4. Divide the leading term \( 28x^2 \) by \( 4x \):
\[
\frac{28x^2}{4x} = 7x
\]
5. Multiply \( 7x \) by \( 4x - 3 \) and subtract:
\[
7x (4x - 3) = 28x^2 - 21x
\]
```
x^2 + 7x
_______________
4x - 3 | 4x^3 + 25x^2 - 13x - 14
-(4x^3 - 3x^2)
_______________
28x^2 - 13x - 14
-(28x^2 - 21x)
_______________
8x - 14
```
### Step 4: Final Division
6. Now bring down the \( -14 \), making it \( 8x - 14 \).
7. Divide the leading term \( 8x \) by \( 4x \):
\[
\frac{8x}{4x} = 2
\]
8. Multiply \( 2 \) by \( 4x - 3 \) and subtract:
\[
2(4x - 3) = 8x - 6
\]
```
x^2 + 7x + 2
_______________
4x - 3 | 4x^3 + 25x^2 - 13x - 14
-(4x^3 - 3x^2)
_______________
28x^2 - 13x - 14
-(28x^2 - 21x)
_______________
8x - 14
-(8x - 6)
_______________
-8
```
### Step 5: Write the Result
The quotient \( q(x) \) is \( x^2 + 7x + 2 \) and the remainder \( r(x) \) is \( -8 \).
Therefore, the result can be expressed as:
\[
q(x) + \frac{r(x)}{b(x)} = x^2 + 7x + 2 - \frac{8}{4x - 3}
\]
### Final Answer
\[
\text{Answer: } x^2 + 7x + 2 - \frac{8}{4x - 3}
\]
### Step 1: Set Up the Long Division
We will divide \( 4x^3 + 25x^2 - 13x - 14 \) by \( 4x - 3 \).
```
_______________
4x - 3 | 4x^3 + 25x^2 - 13x - 14
```
### Step 2: Begin the Long Division
1. Divide the leading term of the dividend \( 4x^3 \) by the leading term of the divisor \( 4x \):
\[
\frac{4x^3}{4x} = x^2
\]
2. Multiply \( x^2 \) by \( 4x - 3 \) and subtract:
\[
x^2 (4x - 3) = 4x^3 - 3x^2
\]
```
x^2
_______________
4x - 3 | 4x^3 + 25x^2 - 13x - 14
-(4x^3 - 3x^2)
_______________
28x^2 - 13x - 14
```
### Step 3: Continue Dividing
3. Now bring down the \( -13x \), making it \( 28x^2 - 13x \).
4. Divide the leading term \( 28x^2 \) by \( 4x \):
\[
\frac{28x^2}{4x} = 7x
\]
5. Multiply \( 7x \) by \( 4x - 3 \) and subtract:
\[
7x (4x - 3) = 28x^2 - 21x
\]
```
x^2 + 7x
_______________
4x - 3 | 4x^3 + 25x^2 - 13x - 14
-(4x^3 - 3x^2)
_______________
28x^2 - 13x - 14
-(28x^2 - 21x)
_______________
8x - 14
```
### Step 4: Final Division
6. Now bring down the \( -14 \), making it \( 8x - 14 \).
7. Divide the leading term \( 8x \) by \( 4x \):
\[
\frac{8x}{4x} = 2
\]
8. Multiply \( 2 \) by \( 4x - 3 \) and subtract:
\[
2(4x - 3) = 8x - 6
\]
```
x^2 + 7x + 2
_______________
4x - 3 | 4x^3 + 25x^2 - 13x - 14
-(4x^3 - 3x^2)
_______________
28x^2 - 13x - 14
-(28x^2 - 21x)
_______________
8x - 14
-(8x - 6)
_______________
-8
```
### Step 5: Write the Result
The quotient \( q(x) \) is \( x^2 + 7x + 2 \) and the remainder \( r(x) \) is \( -8 \).
Therefore, the result can be expressed as:
\[
q(x) + \frac{r(x)}{b(x)} = x^2 + 7x + 2 - \frac{8}{4x - 3}
\]
### Final Answer
\[
\text{Answer: } x^2 + 7x + 2 - \frac{8}{4x - 3}
\]