Answer :
To divide the polynomial [tex]\(126d^2 - 62d - 16\)[/tex] by [tex]\(9d + 2\)[/tex], we perform polynomial long division. Here are the steps:
1. Identify the Leading Terms:
- The leading term of the dividend [tex]\(126d^2 - 62d - 16\)[/tex] is [tex]\(126d^2\)[/tex].
- The leading term of the divisor [tex]\(9d + 2\)[/tex] is [tex]\(9d\)[/tex].
2. Divide Leading Terms:
- Divide the leading term of the dividend by the leading term of the divisor: [tex]\(\frac{126d^2}{9d} = 14d\)[/tex].
- This is the first term of the quotient.
3. Multiply and Subtract:
- Multiply the entire divisor [tex]\(9d + 2\)[/tex] by the term [tex]\(14d\)[/tex] to get: [tex]\( (14d)(9d + 2) = 126d^2 + 28d\)[/tex].
- Subtract this result from the original dividend:
[tex]\[
(126d^2 - 62d - 16) - (126d^2 + 28d) = -90d - 16
\][/tex]
4. Repeat the Process:
- Now take the new polynomial [tex]\(-90d - 16\)[/tex].
- Divide its leading term [tex]\(-90d\)[/tex] by the leading term of the divisor [tex]\(9d\)[/tex]: [tex]\(\frac{-90d}{9d} = -10\)[/tex].
- This is the next term in the quotient.
5. Multiply and Subtract Again:
- Multiply the entire divisor [tex]\(9d + 2\)[/tex] by [tex]\(-10\)[/tex], obtaining: [tex]\(-10(9d + 2) = -90d - 20\)[/tex].
- Subtract this from [tex]\(-90d - 16\)[/tex]:
[tex]\[
(-90d - 16) - (-90d - 20) = 4
\][/tex]
6. Determine the Quotient and Remainder:
- The result of the division is a quotient of [tex]\(14d - 10\)[/tex] with a remainder of 4.
- Represent the remainder as a fraction: [tex]\(\frac{4}{9d + 2}\)[/tex].
Thus, the final answer is:
[tex]\[
14d - 10 + \frac{4}{9d + 2}
\][/tex]
1. Identify the Leading Terms:
- The leading term of the dividend [tex]\(126d^2 - 62d - 16\)[/tex] is [tex]\(126d^2\)[/tex].
- The leading term of the divisor [tex]\(9d + 2\)[/tex] is [tex]\(9d\)[/tex].
2. Divide Leading Terms:
- Divide the leading term of the dividend by the leading term of the divisor: [tex]\(\frac{126d^2}{9d} = 14d\)[/tex].
- This is the first term of the quotient.
3. Multiply and Subtract:
- Multiply the entire divisor [tex]\(9d + 2\)[/tex] by the term [tex]\(14d\)[/tex] to get: [tex]\( (14d)(9d + 2) = 126d^2 + 28d\)[/tex].
- Subtract this result from the original dividend:
[tex]\[
(126d^2 - 62d - 16) - (126d^2 + 28d) = -90d - 16
\][/tex]
4. Repeat the Process:
- Now take the new polynomial [tex]\(-90d - 16\)[/tex].
- Divide its leading term [tex]\(-90d\)[/tex] by the leading term of the divisor [tex]\(9d\)[/tex]: [tex]\(\frac{-90d}{9d} = -10\)[/tex].
- This is the next term in the quotient.
5. Multiply and Subtract Again:
- Multiply the entire divisor [tex]\(9d + 2\)[/tex] by [tex]\(-10\)[/tex], obtaining: [tex]\(-10(9d + 2) = -90d - 20\)[/tex].
- Subtract this from [tex]\(-90d - 16\)[/tex]:
[tex]\[
(-90d - 16) - (-90d - 20) = 4
\][/tex]
6. Determine the Quotient and Remainder:
- The result of the division is a quotient of [tex]\(14d - 10\)[/tex] with a remainder of 4.
- Represent the remainder as a fraction: [tex]\(\frac{4}{9d + 2}\)[/tex].
Thus, the final answer is:
[tex]\[
14d - 10 + \frac{4}{9d + 2}
\][/tex]
