Use the long division method to find the result when 4, x, cubed, plus, 3, x, squared, minus, 30, x, minus, 94x

Question

3
+3x
2
−30x−9 is divided by x, plus, 3x+3.

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

To solve the problem of dividing \( 4x^3 + 3x^2 - 30x - 94 \) by \( x + 3 \) using long division, we proceed as follows:

Step 1: Set up the long division

                   _____________________
x + 3            |   4x^3 + 3x^2 - 30x - 94

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

The leading term of the dividend is \( 4x^3 \) and the leading term of the divisor is \( x \).

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

We will write \( 4x^2 \) above the long division line:

                   4x^2
                   _____________________
x + 3            |   4x^3 + 3x^2 - 30x - 94

Step 3: Multiply the entire divisor by this term and subtract

Now we multiply \( 4x^2 \) by \( x + 3 \):

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

Now, we subtract this result from the original polynomial:

                    4x^2
                   _____________________
x + 3            |   4x^3 + 3x^2 - 30x - 94
                   - (4x^3 + 12x^2)
                   ______________________
                          -9x^2 - 30x - 94

Step 4: Repeat the process for the new polynomial

Now take the new leading term \(-9x^2\) and divide it by the leading term of the divisor \(x\):

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

Write \(-9x\) above the division line:

                   4x^2 - 9x
                   _____________________
x + 3            |   4x^3 + 3x^2 - 30x - 94
                   - (4x^3 + 12x^2)
                   ______________________
                          -9x^2 - 30x - 94
                          - (-9x^2 - 27x)
                   ______________________
                               -3x - 94

Step 5: Do it again for \(-3x\)

Take the new leading term \(-3x\) and divide it by \(x\).

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

Write \(-3\) above the division line:

                   4x^2 - 9x - 3
                   ______________________
x + 3            |   4x^3 + 3x^2 - 30x - 94
                   - (4x^3 + 12x^2)
                   ______________________
                          -9x^2 - 30x - 94
                          - (-9x^2 - 27x)
                   ______________________
                               -3x - 94
                               - (-3x - 9)
                   ______________________
                                   -85

Step 6: Conclusion

The final result of performing the long division is:

\[ \text{Answer: } 4x^2 - 9x - 3 \quad \text{with a remainder of } -85. \]

So, the result when dividing \( 4x^3 + 3x^2 - 30x - 94 \) by \( x + 3 \) is

\[ 4x^2 - 9x - 3 \quad \text{R} -85 \]