Question

Use the long division method to find the result when 9, x, cubed, plus, 24, x, squared, minus, 11, x, plus, 309x
3
+24x
2
−11x+30 is divided by 3, x, plus, 103x+10.





Solve the problem just like the example below
⬇️
Use the long division method to find the result when
4, x, cubed, plus, 12, x, squared, plus, 27, x, plus, 27
4x
3
+12x
2
+27x+27 is divided by
2, x, plus, 3
2x+3.
Use Long Division:
Use Long Division:


2, x, squared
2x
2
plus, 3, x
+3x
plus, 9
+9


2, x
2x
plus, 3
+3
right parenthesis
)
4, x, cubed
4x
3
plus, 12, x, squared
+12x
2
plus, 27, x
+27x
plus, 27
+27



start superscript, minus, end superscript, 4, x, cubed
−
4x
3
plus, start superscript, minus, end superscript, 6, x, squared
+
−
6x
2






6, x, squared
6x
2
plus, 27, x
+27x
plus, 27
+27




start superscript, minus, end superscript, 6, x, squared
−
6x
2
plus, start superscript, minus, end superscript, 9, x
+
−
9x






18, x
18x
plus, 27
+27





start superscript, minus, end superscript, 18, x
−
18x
plus, start superscript, minus, end superscript, 27
+
−
27






0
0

Answer: , 2, x, squared, plus, 3, x, plus, 9
Answer: 2x
2
+3x+9

Answer 1

Let's use the long division method to find the result when \( 9x^3 + 24x^2 - 11x + 30 \) is divided by \( 3x + 10 \).

**Step 1:** Write down the polynomial division in long division format.

```
_________________________
3x + 10 | 9x^3 + 24x^2 - 11x + 30
```

**Step 2:** Determine how many times \( 3x \) goes into \( 9x^3 \).

\( 9x^3 \div 3x = 3x^2 \)

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

```
3x^2
_________________________
3x + 10 | 9x^3 + 24x^2 - 11x + 30
```

**Step 3:** Multiply \( 3x^2 \) by \( 3x + 10 \) and subtract from the original polynomial.

\[
3x^2(3x + 10) = 9x^3 + 30x^2
\]

Now subtract this from the original polynomial:

\[
(9x^3 + 24x^2) - (9x^3 + 30x^2) = -6x^2
\]

Bring down the next term from the original polynomial:

\[
-6x^2 - 11x
\]

Now, the expression is:

```
3x^2
_________________________
3x + 10 | 9x^3 + 24x^2 - 11x + 30
- (9x^3 + 30x^2)
_________________________
-6x^2 - 11x + 30
```

**Step 4:** Now, perform the division again. Determine how many times \( 3x \) goes into \( -6x^2 \).

\( -6x^2 \div 3x = -2x \)

Add \( -2x \) to the quotient:

```
3x^2 - 2x
_________________________
3x + 10 | 9x^3 + 24x^2 - 11x + 30
- (9x^3 + 30x^2)
_________________________
-6x^2 - 11x + 30
- (-6x^2 - 20x)
__________________
9x + 30
```

**Step 5:** Again, divide \( 3x \) into \( 9x \).

\( 9x \div 3x = 3 \)

Add \( 3 \) to the quotient:

```
3x^2 - 2x + 3
_________________________
3x + 10 | 9x^3 + 24x^2 - 11x + 30
- (9x^3 + 30x^2)
_________________________
-6x^2 - 11x + 30
- (-6x^2 - 20x)
__________________
9x + 30
- (9x + 30)
__________________
0
```

**Final Result:**

Since the remainder is \( 0 \), our final answer (the quotient) is:

\[
3x^2 - 2x + 3
\]

**Answer:** \( 3x^2 - 2x + 3 \)