Answer :
To divide the polynomial [tex]\(2x^4 + 15x^3 + 19x^2 - 39x - 45\)[/tex] by [tex]\(x + 5\)[/tex] using synthetic division, follow these steps:
1. Set Up Synthetic Division:
- Write down the coefficients of the polynomial: [tex]\(2, 15, 19, -39, -45\)[/tex].
- Identify the zero of the divisor [tex]\(x + 5\)[/tex], which is [tex]\(-5\)[/tex].
2. Perform the Synthetic Division:
- Bring down the leading coefficient, which is [tex]\(2\)[/tex].
- Multiply the leading coefficient by [tex]\(-5\)[/tex] and add to the next coefficient:
- [tex]\(2 \times (-5) = -10\)[/tex]
- [tex]\(15 + (-10) = 5\)[/tex]
- Multiply the result by [tex]\(-5\)[/tex] and add to the next coefficient:
- [tex]\(5 \times (-5) = -25\)[/tex]
- [tex]\(19 + (-25) = -6\)[/tex]
- Multiply the result by [tex]\(-5\)[/tex] and add to the next coefficient:
- [tex]\(-6 \times (-5) = 30\)[/tex]
- [tex]\(-39 + 30 = -9\)[/tex]
- Multiply the result by [tex]\(-5\)[/tex] and add to the last coefficient:
- [tex]\(-9 \times (-5) = 45\)[/tex]
- [tex]\(-45 + 45 = 0\)[/tex]
3. Result of the Division:
- The coefficients of the quotient are [tex]\(2, 5, -6, -9\)[/tex], corresponding to the polynomial [tex]\(2x^3 + 5x^2 - 6x - 9\)[/tex].
- The remainder is [tex]\(0\)[/tex].
4. Conclusion:
- Since the remainder is [tex]\(0\)[/tex], [tex]\(x + 5\)[/tex] is a factor of [tex]\(2x^4 + 15x^3 + 19x^2 - 39x - 45\)[/tex].
- Thus, the answer to whether [tex]\(x+5\)[/tex] is a factor is yes.
1. Set Up Synthetic Division:
- Write down the coefficients of the polynomial: [tex]\(2, 15, 19, -39, -45\)[/tex].
- Identify the zero of the divisor [tex]\(x + 5\)[/tex], which is [tex]\(-5\)[/tex].
2. Perform the Synthetic Division:
- Bring down the leading coefficient, which is [tex]\(2\)[/tex].
- Multiply the leading coefficient by [tex]\(-5\)[/tex] and add to the next coefficient:
- [tex]\(2 \times (-5) = -10\)[/tex]
- [tex]\(15 + (-10) = 5\)[/tex]
- Multiply the result by [tex]\(-5\)[/tex] and add to the next coefficient:
- [tex]\(5 \times (-5) = -25\)[/tex]
- [tex]\(19 + (-25) = -6\)[/tex]
- Multiply the result by [tex]\(-5\)[/tex] and add to the next coefficient:
- [tex]\(-6 \times (-5) = 30\)[/tex]
- [tex]\(-39 + 30 = -9\)[/tex]
- Multiply the result by [tex]\(-5\)[/tex] and add to the last coefficient:
- [tex]\(-9 \times (-5) = 45\)[/tex]
- [tex]\(-45 + 45 = 0\)[/tex]
3. Result of the Division:
- The coefficients of the quotient are [tex]\(2, 5, -6, -9\)[/tex], corresponding to the polynomial [tex]\(2x^3 + 5x^2 - 6x - 9\)[/tex].
- The remainder is [tex]\(0\)[/tex].
4. Conclusion:
- Since the remainder is [tex]\(0\)[/tex], [tex]\(x + 5\)[/tex] is a factor of [tex]\(2x^4 + 15x^3 + 19x^2 - 39x - 45\)[/tex].
- Thus, the answer to whether [tex]\(x+5\)[/tex] is a factor is yes.
