Answer :
To find the remaining factors of the polynomial [tex]\(8x^4 - 2x^3 - 145x^2 - 216x - 45\)[/tex], given that one factor is [tex]\((x - 5)\)[/tex], we can use polynomial division. Here's how you can proceed:
1. Identify the Polynomial and Known Factor:
- The polynomial is: [tex]\(8x^4 - 2x^3 - 145x^2 - 216x - 45\)[/tex].
- The known factor is: [tex]\((x - 5)\)[/tex].
2. Divide the Polynomial by the Known Factor:
- Perform polynomial division to divide [tex]\(8x^4 - 2x^3 - 145x^2 - 216x - 45\)[/tex] by [tex]\((x - 5)\)[/tex].
- This division process will yield a quotient and potentially a remainder.
3. Check the Remainder:
- If [tex]\((x - 5)\)[/tex] is indeed a factor, the remainder after division should be 0. This confirms that the division is exact.
4. Find the Quotient:
- After division, the quotient you obtain is [tex]\(8x^3 + 38x^2 + 45x + 9\)[/tex].
- This quotient is the remaining factor of the original polynomial after factoring out [tex]\((x - 5)\)[/tex].
Therefore, the factorization of the polynomial [tex]\(8x^4 - 2x^3 - 145x^2 - 216x - 45\)[/tex] is [tex]\((x - 5)(8x^3 + 38x^2 + 45x + 9)\)[/tex].
This means that once you know [tex]\((x - 5)\)[/tex] is a factor, the other factors can be found by considering the quotient from the polynomial division.
1. Identify the Polynomial and Known Factor:
- The polynomial is: [tex]\(8x^4 - 2x^3 - 145x^2 - 216x - 45\)[/tex].
- The known factor is: [tex]\((x - 5)\)[/tex].
2. Divide the Polynomial by the Known Factor:
- Perform polynomial division to divide [tex]\(8x^4 - 2x^3 - 145x^2 - 216x - 45\)[/tex] by [tex]\((x - 5)\)[/tex].
- This division process will yield a quotient and potentially a remainder.
3. Check the Remainder:
- If [tex]\((x - 5)\)[/tex] is indeed a factor, the remainder after division should be 0. This confirms that the division is exact.
4. Find the Quotient:
- After division, the quotient you obtain is [tex]\(8x^3 + 38x^2 + 45x + 9\)[/tex].
- This quotient is the remaining factor of the original polynomial after factoring out [tex]\((x - 5)\)[/tex].
Therefore, the factorization of the polynomial [tex]\(8x^4 - 2x^3 - 145x^2 - 216x - 45\)[/tex] is [tex]\((x - 5)(8x^3 + 38x^2 + 45x + 9)\)[/tex].
This means that once you know [tex]\((x - 5)\)[/tex] is a factor, the other factors can be found by considering the quotient from the polynomial division.
