Question

Use the long division method to find the result when 4, x, cubed, plus, 12, x, squared, plus, 19, x, plus, 74x
3
+12x
2
+19x+7 is divided by 2, x, plus, 12x+1.





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

Let's divide the polynomial \( 4x^3 + 12x^2 + 19x + 7 \) by \( 2x + 12 \) using long division.

**Use Long Division:**

\[
\begin{array}{r|l}
2x + 12 & 4x^3 + 12x^2 + 19x + 7 \\
\hline
& 2x^2 \\
\end{array}
\]

1. **First Division**:
- Divide \( 4x^3 \) by \( 2x \) to get \( 2x^2 \).

Write \( 2x^2 \) above the division line.

\[
\begin{array}{r|l}
2x + 12 & 4x^3 + 12x^2 + 19x + 7 \\
\hline
& 2x^2 \\
2x^2 \cdot (2x + 12) = 4x^3 + 24x^2 \\
\end{array}
\]

2. **Subtract**:
- Subtract \( (4x^3 + 24x^2) \) from \( (4x^3 + 12x^2) \):

\[
(4x^3 + 12x^2 + 19x + 7) - (4x^3 + 24x^2) = -12x^2 + 19x + 7
\]

Now the new expression is \( -12x^2 + 19x + 7 \):

\[
\begin{array}{r|l}
2x + 12 & 4x^3 + 12x^2 + 19x + 7 \\
\hline
& 2x^2 \\
& - (4x^3 + 24x^2) \\
\hline
& -12x^2 + 19x + 7 \\
\end{array}
\]

3. **Second Division**:
- Divide \( -12x^2 \) by \( 2x \) to get \( -6x \).

Write \( -6x \) above the division line.

\[
\begin{array}{r|l}
2x + 12 & 4x^3 + 12x^2 + 19x + 7 \\
\hline
& 2x^2 - 6x \\
-6x \cdot (2x + 12) = -12x^2 - 72x \\
\end{array}
\]

4. **Subtract**:
- Subtract \( (-12x^2 - 72x) \) from \( (-12x^2 + 19x + 7) \):

\[
(-12x^2 + 19x + 7) - (-12x^2 - 72x) = 91x + 7
\]

Now the new expression is \( 91x + 7 \):

\[
\begin{array}{r|l}
2x + 12 & 4x^3 + 12x^2 + 19x + 7 \\
\hline
& 2x^2 - 6x \\
& - (4x^3 + 24x^2) \\
& - (-12x^2 + 72x) \\
\hline
& 91x + 7 \\
\end{array}
\]

5. **Third Division**:
- Divide \( 91x \) by \( 2x \) to get \( 45.5 \).

Write \( 45.5 \) above the division line.

\[
\begin{array}{r|l}
2x + 12 & 4x^3 + 12x^2 + 19x + 7 \\
\hline
& 2x^2 - 6x + 45.5 \\
45.5 \cdot (2x + 12) = 91x + 546 \\
\end{array}
\]

6. **Subtract**:
- Subtract \( (91x + 546) \) from \( (91x + 7) \):

\[
(91x + 7) - (91x + 546) = -539
\]

Now the new expression is \( -539 \):

\[
\begin{array}{r|l}
2x + 12 & 4x^3 + 12x^2 + 19x + 7 \\
\hline
& 2x^2 - 6x + 45.5 \\
& - (91x + 546) \\
\hline
& -539 \\
\end{array}
\]

So the final result is:

**Answer**: \( 2x^2 - 6x + 45.5 \) with a remainder of \( -539 \).