Sure! Let's work through the division of the polynomial [tex]\( \frac{2 x^4 - 3 x^3 + a x^2 + b x + 2 b}{x^2 - 2 x - 3} \)[/tex] step-by-step.
#### Step 1: Set Up the Polynomial Division
We need to divide the polynomial [tex]\( 2 x^4 - 3 x^3 + a x^2 + b x + 2 b \)[/tex] by [tex]\( x^2 - 2 x - 3 \)[/tex]. This is analogous to long division with numbers but here we are dealing with polynomials.
#### Step 2: Division Process
1. First Term:
- Divide the leading term of the numerator [tex]\( 2 x^4 \)[/tex] by the leading term of the denominator [tex]\( x^2 \)[/tex]:
[tex]\[
\frac{2 x^4}{x^2} = 2 x^2
\][/tex]
- Multiply [tex]\( 2 x^2 \)[/tex] by the entire denominator:
[tex]\[
2 x^2 \cdot (x^2 - 2 x - 3) = 2 x^4 - 4 x^3 - 6 x^2
\][/tex]
- Subtract this product from the original polynomial:
[tex]\[
(2 x^4 - 3 x^3 + a x^2 + b x + 2 b) - (2 x^4 - 4 x^3 - 6 x^2) = x^3 + (a + 6) x^2 + b x + 2 b
\][/tex]
2. Second Term:
- Divide the leading term of the new polynomial [tex]\( x^3 \)[/tex] by the leading term of the denominator [tex]\( x^2 \)[/tex]:
[tex]\[
\frac{x^3}{x^2} = x
\][/tex]
- Multiply [tex]\( x \)[/tex] by the entire denominator:
[tex]\[
x \cdot (x^2 - 2 x - 3) = x^3 - 2 x^2 - 3 x
\][/tex]
- Subtract this product from the new polynomial:
[tex]\[
(x^3 + (a + 6) x^2 + b x + 2 b) - (x^3 - 2 x^2 - 3 x) = (a + 8) x^2 + (b + 3) x + 2 b
\][/tex]
3. Third Term:
- Divide the leading term of the new polynomial [tex]\( (a + 8) x^2 \)[/tex] by the leading term of the denominator [tex]\( x^2 \)[/tex]:
[tex]\[
\frac{(a + 8) x^2}{x^2} = a + 8
\][/tex]
- Multiply [tex]\( (a + 8) \)[/tex] by the entire denominator:
[tex]\[
(a + 8) \cdot (x^2 - 2 x - 3) = a x^2 + 8 x^2 - 2 a x - 16 x - 3 a - 24
\][/tex]
- Subtract this product from the new polynomial:
[tex]\[
(a + 8) x^2 + (b + 3) x + 2 b - (a x^2 + 8 x^2 - 2 a x - 16 x - 3 a - 24) = (3 a + 2 b) + (2 a + b + 19) x + 24
\][/tex]
#### Step 3: Compiling the Results
So, the quotient from the division is:
[tex]\[
2 x^2 + x + a + 8
\][/tex]
And the remainder is:
[tex]\[
(3 a + 2 b) + (2 a + b + 19) x + 24
\][/tex]
Thus, the final result of the polynomial division is:
[tex]\[
\frac{2 x^4 - 3 x^3 + a x^2 + b x + 2 b}{x^2 - 2 x - 3} = (2 x^2 + x + a + 8) + \frac{(3 a + 2 b) + (2 a + b + 19) x + 24}{x^2 - 2 x - 3}
\][/tex]
Putting it all together, we get:
Quotient: [tex]\(2 x^2 + x + a + 8\)[/tex]
Remainder: [tex]\(3a + 2b + x(2a + b + 19) + 24\)[/tex]
