Use the long division method to find the result when 4, x, cubed, plus, 12, x, squared, plus, 9, x, plus, 24x

Question

3
+12x
2
+9x+2 is divided by x, plus, 2x+2.

Solve the problem just like the example below
⬇️
Use the long division method to find the result when
4, x, cubed, plus, 5, x, squared, plus, 5, x, plus, 1
4x
3
+5x
2
+5x+1 is divided by
4, x, plus, 1
4x+1.
Use Long Division:
Use Long Division:

x, squared
x
2
plus, x
+x
plus, 1
+1

4, x
4x
plus, 1
+1
right parenthesis
)
4, x, cubed
4x
3
plus, 5, x, squared
+5x
2
plus, 5, x
+5x
plus, 1
+1

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

4, x, squared
4x
2
plus, 5, x
+5x
plus, 1
+1

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

4, x
4x
plus, 1
+1

start superscript, minus, end superscript, 4, x
−
4x
plus, start superscript, minus, end superscript, 1
+
−
1

0
0

Answer: , x, squared, plus, x, plus, 1
Answer: x
2
+x+1

Answer 1

To divide \(4x^3 + 12x^2 + 9x + 24\) by \(x + 2\) using long division, follow the steps below:

Step 1: Set up the Long Division

                ______________________
x + 2      |  4x^3 + 12x^2 + 9x + 24

Step 2: Divide the Leading Terms

Divide the leading term of the dividend \(4x^3\) by the leading term of the divisor \(x\):

\[ \frac{4x^3}{x} = 4x^2 \] Now, we'll write \(4x^2\) above the division bar.

Step 3: Multiply and Subtract

Multiply \(4x^2\) by the entire divisor \(x + 2\):

\[ 4x^2(x + 2) = 4x^3 + 8x^2 \] Now, write this underneath the dividend and subtract it:

                4x^2
               _______________
x + 2      |  4x^3 + 12x^2 + 9x + 24
               -(4x^3 + 8x^2)
               ___________________
                        4x^2 + 9x + 24

Step 4: Bring Down and Repeat

Now bring down the next term (which is already present) and continue:

Divide the new leading term \(4x^2\) by \(x\):

\[ \frac{4x^2}{x} = 4x \] Write \(4x\) above the division bar.

Multiply \(4x\) by the entire divisor \(x + 2\):

\[ 4x(x + 2) = 4x^2 + 8x \] Subtract this from \(4x^2 + 9x + 24\):

                4x^2 + 4x
               _______________
x + 2      |  4x^3 + 12x^2 + 9x + 24
               -(4x^3 + 8x^2)
               ___________________
                        4x^2 + 9x + 24
                       -(4x^2 + 8x)
                       _______________
                               x + 24

Step 5: Divide Again

Now divide \(x\) by \(x\):

\[ \frac{x}{x} = 1 \] Write \(1\) above the division bar.

Multiply \(1\) by the divisor \(x + 2\):

\[ 1(x + 2) = x + 2 \]

Subtract this from \(x + 24\):

                4x^2 + 4x + 1
               _________________
x + 2      |  4x^3 + 12x^2 + 9x + 24
               -(4x^3 + 8x^2)
               ___________________
                        4x^2 + 9x + 24
                       -(4x^2 + 8x)
                       _______________
                               x + 24
                              -(x + 2)
                             _______________
                                   22

Step 6: Write the Final Result

After performing the long division, we are left with a remainder of \(22\). Therefore, we can express our division as:

\[ 4x^3 + 12x^2 + 9x + 24 = (x + 2)(4x^2 + 4x + 1) + 22 \]

Answer

The quotient is \(4x^2 + 4x + 1\) and the remainder is \(22\).

Final Answer: \(4x^2 + 4x + 1\) with a remainder of \(22\).