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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

1 answer

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.

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

  1. 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 \( -12x^2 + 19x + 7 \):

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

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

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

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

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

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

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

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

  1. Third Division:
    • Divide \( 91x \) by \( 2x \) to get \( 45.5 \).

Write \( 45.5 \) above the division line.

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

  1. 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 \( -539 \).