Answer :
To solve the division [tex]\((-10c^5 + 89c^4 + 74c^2 + 62c + 3) \div (c - 9)\)[/tex], we'll perform polynomial long division. Here's a step-by-step breakdown:
1. Divide the leading terms:
- Divide [tex]\(-10c^5\)[/tex] by [tex]\(c\)[/tex], which gives [tex]\(-10c^4\)[/tex].
2. Multiply and subtract:
- Multiply [tex]\(-10c^4\)[/tex] by [tex]\(c - 9\)[/tex], yielding [tex]\(-10c^5 + 90c^4\)[/tex].
- Subtract this result from the original polynomial:
[tex]\[
(-10c^5 + 89c^4) - (-10c^5 + 90c^4) = -c^4
\][/tex]
3. Bring down the next term and repeat:
- Bring down [tex]\(+ 74c^2\)[/tex], so now it's [tex]\(-c^4 + 74c^2\)[/tex].
- Divide [tex]\(-c^4\)[/tex] by [tex]\(c\)[/tex], resulting in [tex]\(-c^3\)[/tex].
- Multiply [tex]\(-c^3\)[/tex] by [tex]\(c - 9\)[/tex], getting [tex]\(-c^4 + 9c^3\)[/tex].
- Subtract:
[tex]\[
(-c^4) - (-c^4 + 9c^3) = -9c^3
\][/tex]
4. Continue the process:
- Bring down [tex]\(62c\)[/tex], making it [tex]\(-9c^3 + 62c\)[/tex].
- Divide [tex]\(-9c^3\)[/tex] by [tex]\(c\)[/tex] to get [tex]\(-9c^2\)[/tex].
- Multiply [tex]\(-9c^2\)[/tex] by [tex]\(c - 9\)[/tex], resulting in [tex]\(-9c^3 + 81c^2\)[/tex].
- Subtract:
[tex]\[
(-9c^3 + 74c^2) - (-9c^3 + 81c^2) = -7c^2
\][/tex]
5. Proceed further:
- Bring down next term [tex]\(+62c\)[/tex], resulting in [tex]\(-7c^2 + 62c\)[/tex].
- Divide [tex]\(-7c^2\)[/tex] by [tex]\(c\)[/tex] to get [tex]\(-7c\)[/tex].
- Multiply [tex]\(-7c\)[/tex] by [tex]\(c - 9\)[/tex], giving [tex]\(-7c^2 + 63c\)[/tex].
- Subtract:
[tex]\[
(-7c^2 + 62c) - (-7c^2 + 63c) = -c
\][/tex]
6. Last steps:
- Bring down [tex]\(+3\)[/tex], making it [tex]\(-c + 3\)[/tex].
- Divide [tex]\(-c\)[/tex] by [tex]\(c\)[/tex] to get [tex]\(-1\)[/tex].
- Multiply [tex]\(-1\)[/tex] by [tex]\(c - 9\)[/tex], resulting in [tex]\(-c + 9\)[/tex].
- Subtract:
[tex]\[
(-c + 3) - (-c + 9) = -6
\][/tex]
The quotient is [tex]\(-10c^4 - c^3 - 9c^2 - 7c - 1\)[/tex], with a remainder of [tex]\(-6\)[/tex].
Therefore, the polynomial division results in:
[tex]\[
(-10c^5 + 89c^4 + 74c^2 + 62c + 3) \div (c-9) = -10c^4 - c^3 - 9c^2 - 7c - 1 \quad \text{with a remainder of} \quad -6.
\][/tex]
1. Divide the leading terms:
- Divide [tex]\(-10c^5\)[/tex] by [tex]\(c\)[/tex], which gives [tex]\(-10c^4\)[/tex].
2. Multiply and subtract:
- Multiply [tex]\(-10c^4\)[/tex] by [tex]\(c - 9\)[/tex], yielding [tex]\(-10c^5 + 90c^4\)[/tex].
- Subtract this result from the original polynomial:
[tex]\[
(-10c^5 + 89c^4) - (-10c^5 + 90c^4) = -c^4
\][/tex]
3. Bring down the next term and repeat:
- Bring down [tex]\(+ 74c^2\)[/tex], so now it's [tex]\(-c^4 + 74c^2\)[/tex].
- Divide [tex]\(-c^4\)[/tex] by [tex]\(c\)[/tex], resulting in [tex]\(-c^3\)[/tex].
- Multiply [tex]\(-c^3\)[/tex] by [tex]\(c - 9\)[/tex], getting [tex]\(-c^4 + 9c^3\)[/tex].
- Subtract:
[tex]\[
(-c^4) - (-c^4 + 9c^3) = -9c^3
\][/tex]
4. Continue the process:
- Bring down [tex]\(62c\)[/tex], making it [tex]\(-9c^3 + 62c\)[/tex].
- Divide [tex]\(-9c^3\)[/tex] by [tex]\(c\)[/tex] to get [tex]\(-9c^2\)[/tex].
- Multiply [tex]\(-9c^2\)[/tex] by [tex]\(c - 9\)[/tex], resulting in [tex]\(-9c^3 + 81c^2\)[/tex].
- Subtract:
[tex]\[
(-9c^3 + 74c^2) - (-9c^3 + 81c^2) = -7c^2
\][/tex]
5. Proceed further:
- Bring down next term [tex]\(+62c\)[/tex], resulting in [tex]\(-7c^2 + 62c\)[/tex].
- Divide [tex]\(-7c^2\)[/tex] by [tex]\(c\)[/tex] to get [tex]\(-7c\)[/tex].
- Multiply [tex]\(-7c\)[/tex] by [tex]\(c - 9\)[/tex], giving [tex]\(-7c^2 + 63c\)[/tex].
- Subtract:
[tex]\[
(-7c^2 + 62c) - (-7c^2 + 63c) = -c
\][/tex]
6. Last steps:
- Bring down [tex]\(+3\)[/tex], making it [tex]\(-c + 3\)[/tex].
- Divide [tex]\(-c\)[/tex] by [tex]\(c\)[/tex] to get [tex]\(-1\)[/tex].
- Multiply [tex]\(-1\)[/tex] by [tex]\(c - 9\)[/tex], resulting in [tex]\(-c + 9\)[/tex].
- Subtract:
[tex]\[
(-c + 3) - (-c + 9) = -6
\][/tex]
The quotient is [tex]\(-10c^4 - c^3 - 9c^2 - 7c - 1\)[/tex], with a remainder of [tex]\(-6\)[/tex].
Therefore, the polynomial division results in:
[tex]\[
(-10c^5 + 89c^4 + 74c^2 + 62c + 3) \div (c-9) = -10c^4 - c^3 - 9c^2 - 7c - 1 \quad \text{with a remainder of} \quad -6.
\][/tex]
