How to Use Synthetic Division

Oct 13, 2025 by pynkocean.com 2 views

**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.