Use the long division method to find the result when 9, x, cubed, plus, 30, x, squared, plus, 30, x, plus, 129x

Question

3
+30x
2
+30x+12 is divided by 3, x, plus, 43x+4. 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 use the long division method to divide \(9x^3 + 30x^2 + 30x + 12\) by \(3x + 4\).

Step 1: Set up the long division

            __________________________
3x + 4    |  9x^3 + 30x^2 + 30x + 12

Step 2: Divide the leading terms

Divide \(9x^3\) by \(3x\):

\[ \frac{9x^3}{3x} = 3x^2 \]

Step 3: Multiply and subtract

Multiply \(3x^2\) by \(3x + 4\):

\[ 3x^2(3x + 4) = 9x^3 + 12x^2 \]

Now subtract:

\[ (9x^3 + 30x^2 + 30x + 12) - (9x^3 + 12x^2) = 18x^2 + 30x + 12 \]

Now our division looks like this:

            3x^2
            __________________________
3x + 4    |  9x^3 + 30x^2 + 30x + 12
           - (9x^3 + 12x^2)
           __________________________
                   18x^2 + 30x + 12

Step 4: Repeat the process

Now divide the leading term \(18x^2\) by \(3x\):

\[ \frac{18x^2}{3x} = 6x \]

Multiply \(6x\) by \(3x + 4\):

\[ 6x(3x + 4) = 18x^2 + 24x \]

Now subtract:

\[ (18x^2 + 30x + 12) - (18x^2 + 24x) = 6x + 12 \]

Now our long division looks like this:

            3x^2 + 6x
            __________________________
3x + 4    |  9x^3 + 30x^2 + 30x + 12
           - (9x^3 + 12x^2)
           __________________________
                   18x^2 + 30x + 12
                   - (18x^2 + 24x)
                   __________________________
                           6x + 12

Step 5: Continue the process

Now divide the leading term \(6x\) by \(3x\):

\[ \frac{6x}{3x} = 2 \]

Multiply \(2\) by \(3x + 4\):

\[ 2(3x + 4) = 6x + 8 \]

Now subtract:

\[ (6x + 12) - (6x + 8) = 4 \]

Final result of the long division

Now our division shows:

            3x^2 + 6x + 2
            __________________________
3x + 4    |  9x^3 + 30x^2 + 30x + 12
           - (9x^3 + 12x^2)
           __________________________
                   18x^2 + 30x + 12
                   - (18x^2 + 24x)
                   __________________________
                           6x + 12
                           - (6x + 8)
                           __________________________
                                   4

Step 6: Write in the required form

The quotient \(q(x)\) is \(3x^2 + 6x + 2\), and the remainder \(r(x)\) is \(4\).

Thus, the result can be expressed as:

\[ 3x^2 + 6x + 2 + \frac{4}{3x + 4} \]

Final answer:

\[ \boxed{3x^2 + 6x + 2 + \frac{4}{3x + 4}} \]