Problem:
Perform the long algebraic division of \( -2x^4 - 4x^3 - 5x^2 + 2x + 3 \) by \( x^2 + 2x + 3 \).
Solution:
We are tasked with dividing the polynomial \( -2x^4 - 4x^3 - 5x^2 + 2x + 3 \) by \( x^2 + 2x + 3 \). Let's go through the steps of polynomial long division.
#### Step 1: Set up the division
Write the dividend \( -2x^4 - 4x^3 - 5x^2 + 2x + 3 \) and the divisor \( x^2 + 2x + 3 \).
\[
\require{enclose}
\begin{array}{r|rrrrr}
x^2 + 2x + 3 & -2x^4 & -4x^3 & -5x^2 & +2x & +3 \\
\end{array}
\]
#### Step 2: Divide the leading term of the dividend by the leading term of the divisor
The leading term of the dividend is \( -2x^4 \), and the leading term of the divisor is \( x^2 \). Divide these:
\[
\frac{-2x^4}{x^2} = -2x^2
\]
So, the first term of the quotient is \( -2x^2 \).
#### Step 3: Multiply the entire divisor by this term and subtract from the dividend
Multiply \( -2x^2 \) by the divisor \( x^2 + 2x + 3 \):
\[
-2x^2 \cdot (x^2 + 2x + 3) = -2x^4 - 4x^3 - 6x^2
\]
Subtract this from the original dividend:
\[
\begin{array}{r}
-2x^4 - 4x^3 - 5x^2 + 2x + 3 \\
\underline{-(-2x^4 - 4x^3 - 6x^2)} \\
0x^4 + 0x^3 + x^2 + 2x + 3
\end{array}
\]
The result is \( x^2 + 2x + 3 \).
#### Step 4: Repeat the process with the new polynomial
Now, divide the leading term of the new polynomial \( x^2 \) by the leading term of the divisor \( x^2 \):
\[
\frac{x^2}{x^2} = 1
\]
So, the next term of the quotient is \( 1 \).
#### Step 5: Multiply the entire divisor by this term and subtract
Multiply \( 1 \) by the divisor \( x^2 + 2x + 3 \):
\[
1 \cdot (x^2 + 2x + 3) = x^2 + 2x + 3
\]
Subtract this from the current polynomial:
\[
\begin{array}{r}
x^2 + 2x + 3 \\
\underline{-(x^2 + 2x + 3)} \\
0
\end{array}
\]
The remainder is \( 0 \).
#### Step 6: Write the final quotient
The quotient is the sum of the terms we found: \( -2x^2 + 1 \).
Thus, the result of the division is:
\[
\frac{-2x^4 - 4x^3 - 5x^2 + 2x + 3}{x^2 + 2x + 3} = -2x^2 + 1
\]
Final Answer:
\[
\boxed{-2x^2 + 1}
\]
Parent Tip: Review the logic above to help your child master the concept of algebraic long division worksheet pdf.
