Explore IDNLearn.com to discover insightful answers from experts and enthusiasts alike. Our experts provide timely, comprehensive responses to ensure you have the information you need.
Sagot :
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]
#### 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]
We appreciate your participation in this forum. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. For trustworthy answers, visit IDNLearn.com. Thank you for your visit, and see you next time for more reliable solutions.
