Question

Use the long division method to find the result when 12, x, cubed, plus, 8, x, squared, minus, 7, x, minus, 912x
3
+8x
2
−7x−9 is divided by 3, x, minus, 13x−1. 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
​

Answer 1

Let's perform the long division of the polynomial \(12x^3 + 8x^2 - 7x - 912\) by the polynomial \(3x - 13\).

**Step 1:** Set up the long division.

```
_________
3x - 13 | 12x^3 + 8x^2 - 7x - 912
```

**Step 2:** Divide the leading term of the dividend by the leading term of the divisor.

- Divide \(12x^3\) by \(3x\):

\[
\frac{12x^3}{3x} = 4x^2
\]

Write \(4x^2\) above the division line.

```
4x^2
_________
3x - 13 | 12x^3 + 8x^2 - 7x - 912
```

**Step 3:** Multiply the entire divisor \(3x - 13\) by \(4x^2\) and write the result below the dividend.

\[
4x^2 \cdot (3x - 13) = 12x^3 - 52x^2
\]

Now write this below the corresponding terms in the dividend.

```
4x^2
_________
3x - 13 | 12x^3 + 8x^2 - 7x - 912
- (12x^3 - 52x^2)
_____________________
60x^2 - 7x - 912
```

**Step 4:** Subtract this from the dividend.

```
60x^2 - 7x - 912
```

**Step 5:** Bring down the next term (here you already have it).

**Step 6:** Divide again. Divide \(60x^2\) by \(3x\):

\[
\frac{60x^2}{3x} = 20x
\]

Write \(20x\) above the division line next to \(4x^2\).

```
4x^2 + 20x
_________
3x - 13 | 12x^3 + 8x^2 - 7x - 912
- (12x^3 - 52x^2)
_____________________
60x^2 - 7x - 912
- (60x^2 - 260x)
_____________________
253x - 912
```

**Step 7:** Subtract this from the previous result.

**Step 8:** Now divide \(253x\) by \(3x\):

\[
\frac{253x}{3x} = \frac{253}{3}
\]

Write \(\frac{253}{3}\) above.

```
4x^2 + 20x + \frac{253}{3}
_________
3x - 13 | 12x^3 + 8x^2 - 7x - 912
- (12x^3 - 52x^2)
_____________________
60x^2 - 7x - 912
- (60x^2 - 260x)
_____________________
253x - 912
- (253x - \frac{3289}{3})
_____________________
-912 + \frac{3289}{3} = \frac{-2736 + 3289}{3} = \frac{553}{3}
```

**Step 9:** Write the final expression.

The quotient \(q(x)\) is

\[
q(x) = 4x^2 + 20x + \frac{253}{3}
\]

The remainder \(r(x)\) is

\[
r(x) = \frac{553}{3}
\]

The divisor \(b(x)\) is

\[
b(x) = 3x - 13
\]

**Final Answer:**

\[
q(x) + \frac{r(x)}{b(x)} = 4x^2 + 20x + \frac{253}{3} + \frac{\frac{553}{3}}{3x - 13}
\]

Thus, the full result can be expressed as:

\[
4x^2 + 20x + \frac{253}{3} + \frac{553}{3(3x - 13)}
\] or

\[
4x^2 + 20x + \frac{253}{3} + \frac{553}{3(3x - 13)}
\]
is the answer, expressed as \( q(x) + \frac{r(x)}{b(x)} \) form.