Use the long division method to find the result when 9, x, cubed, plus, 24, x, squared, minus, 11, x, plus, 309x

Question

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