Use Synthetic Division To Solve \[$(2x^3 + 4x^2 - 35x + 15) \div (x - 3)\$\]. What Is The Quotient?A. \[$2x^2 - 2x - 29 + \frac{102}{x + 3}\$\]B. \[$2x^2 - 2x - 29 + \frac{102}{x - 3}\$\]C. \[$2x^3 + 10x^2 - 5x\$\]D.
**Use Synthetic Division to Solve a Polynomial Equation** ===========================================================
What is Synthetic Division?
Synthetic division is a shortcut method for dividing a polynomial by a linear factor of the form (x - c). It is a simplified version of the long division method and is used to find the quotient and remainder of a polynomial division. This method is particularly useful when dividing polynomials with a large number of terms.
How to Use Synthetic Division
To use synthetic division, you need to follow these steps:
- Write down the coefficients of the polynomial in a row, with the constant term on the right.
- Write down the value of c, which is the value that x is being divided by.
- Bring down the first coefficient.
- Multiply the value of c by the first coefficient and write the result below the second coefficient.
- Add the second coefficient and the result from step 4.
- Multiply the value of c by the result from step 5 and write the result below the third coefficient.
- Add the third coefficient and the result from step 6.
- Repeat steps 6 and 7 until you have used up all the coefficients.
- The final result is the quotient and remainder.
Example: Use Synthetic Division to Solve (2x^3 + 4x^2 - 35x + 15) ÷ (x - 3)
To solve this problem, we will use synthetic division with c = 3.
| 3 | 2 4 -35 15 |
|---|---|
| 2 | 6 -30 -60 45 |
| 2 | 12 -60 -120 90 |
| 2 | 24 -120 -240 180 |
| 2 | 50 -250 -500 450 |
The final result is:
Quotient: 2x^2 - 2x - 29 Remainder: 102
Quotient: 2x^2 - 2x - 29 + \frac{102}{x + 3}
Therefore, the quotient is 2x^2 - 2x - 29 + \frac{102}{x + 3}.
Q&A
Q: What is the purpose of synthetic division? A: The purpose of synthetic division is to find the quotient and remainder of a polynomial division.
Q: How do I use synthetic division? A: To use synthetic division, you need to follow these steps: write down the coefficients of the polynomial in a row, with the constant term on the right; write down the value of c, which is the value that x is being divided by; bring down the first coefficient; multiply the value of c by the first coefficient and write the result below the second coefficient; add the second coefficient and the result from step 4; multiply the value of c by the result from step 5 and write the result below the third coefficient; add the third coefficient and the result from step 6; repeat steps 6 and 7 until you have used up all the coefficients; the final result is the quotient and remainder.
Q: What is the quotient of (2x^3 + 4x^2 - 35x + 15) ÷ (x - 3)? A: The quotient of (2x^3 + 4x^2 - 35x + 15) ÷ (x - 3) is 2x^2 - 2x - 29 + \frac{102}{x + 3}.
Q: What is the remainder of (2x^3 + 4x^2 - 35x + 15) ÷ (x - 3)? A: The remainder of (2x^3 + 4x^2 - 35x + 15) ÷ (x - 3) is 102.
Q: Can I use synthetic division to divide a polynomial by a quadratic factor? A: No, synthetic division is only used to divide a polynomial by a linear factor of the form (x - c). To divide a polynomial by a quadratic factor, you need to use a different method, such as long division or factoring.
Q: Can I use synthetic division to divide a polynomial by a rational factor? A: No, synthetic division is only used to divide a polynomial by a linear factor of the form (x - c). To divide a polynomial by a rational factor, you need to use a different method, such as long division or factoring.
Q: Can I use synthetic division to divide a polynomial by a polynomial of degree 0? A: Yes, synthetic division can be used to divide a polynomial by a polynomial of degree 0, which is a constant. In this case, the quotient is the polynomial itself, and the remainder is the constant.
Q: Can I use synthetic division to divide a polynomial by a polynomial of degree 1? A: Yes, synthetic division can be used to divide a polynomial by a polynomial of degree 1, which is a linear factor of the form (x - c). In this case, the quotient is the polynomial divided by the linear factor, and the remainder is the constant.
Q: Can I use synthetic division to divide a polynomial by a polynomial of degree 2 or higher? A: No, synthetic division is only used to divide a polynomial by a linear factor of the form (x - c). To divide a polynomial by a polynomial of degree 2 or higher, you need to use a different method, such as long division or factoring.