Synthetic Division of Polynomials Practice Problems (original) (raw)
Last Updated : 23 Jul, 2025
**Synthetic division is a simplified method used to divide polynomials, particularly useful when the divisor is a linear polynomial. It is quicker and less error-prone compared to the traditional long division of polynomials. This method is advantageous because it involves fewer steps and can be performed without variables, making it an efficient way to handle polynomial division.
In this article, we are going to discuss the synthetic division of polynomials practice problems.
Table of Content
- What is the Synthetic Division of Polynomials?
- How to Perform Synthetic Division of Polynomials?
- Synthetic Division of Polynomial Solved Examples
- Probelms on Synthetic Division of Polynomials : Unsolved
**What is Synthetic Division of Polynomials?
Synthetic division is a simplified method of dividing polynomials which is a **quicker and more efficient alternative to long division, particularly useful for dividing the polynomial by a linear factor of the form x − c. It is **generally used to find out the roots of polynomials and not for the division of factors.
Mathematically, it can be represented as follows:
**P(x)/(x-a) = Q(x) +[R/(x-a)]
Here, Q(x) is quotient polynomial P(x) having linear factor (x-a) and R is the remainder, which is a constant term.
In synthetic division, **we first set the denominator equal to zero to find the number to place in the division box. We then arrange the numerator in descending order, filling in any missing terms with zeros. The coefficients of the polynomial are used in the division process.
**Note: We can perform the synthetic division method, only if the divisor is a linear factor.
Steps for Synthetic Division of Polynomials
Synthetic division of polynomials can be easily performed using a few easy steps:
Suppose we have to divide a polynomial of the form ax2 + bx + c = 0 by its linear factor x - a then,
- **Step 1: List out coefficients of each algebraic term of the polynomial ax2 + bx + c = 0.
- **Step 2: List the zeroes or the roots of the linear factor x - a.
- **Step 3: Write the constant from the divisor x − c to the left.
- **Step 4: Bring down the first coefficient.
- **Step 5: Multiply the constant by the value below the line and write the result under the next coefficient.
- **Step 6: Add the column of numbers and write the result below the line.
- **Step 7: Repeat the multiply and add steps for all coefficients.
- **Step 8: The final row of numbers gives the coefficients of the quotient polynomial, with the last number being the remainder.
**Synthetic Division of Polynomial Practice Problems with Solutions
These Synthetic Division of Polynomials Practice Problems are designed to help you master this efficient method of dividing polynomials:
**1. Divide 2x 3 **+ 3x 2 **+ 5x + 6 by x-2.
- Set up the synthetic division with the divisor x-2 which gives a root of 2.
- Write the coefficients of the dividend: 2, 3, -5, 6.
- Perform synthetic division:
**So, 2x 3 +3x 2 +5x+6 divided by x-2 gives 2x 2 +7x+9 with a remainder of 24 as the final answer.
**2. Divide 4x 3- 6x 2 +x-8 by x+1.
- The divisor is x+1 has a root of -1.
- Write the coefficients of the dividend: 4, -6, 1, -8.
- Perform synthetic division:
**Thus 4x 3 -6x 2 +x-8 divided by x+1 gives the result as 4x 2 -10x+11 with -19 as a remainder.
**3. Divide -3x 3 +4x 2 -2x+5 by x-1.
- The divisor x-1 has a root of 1
- Write the coefficients of the dividend: 1, -3, 4, -2, 5.
- Perform synthetic division:
**Thus -3x 3 +4x 2 -2x+5 divided by x-1 gives x 3 -2x 2 +2x+0 with a remainder of 5.
**4. Divide 5x 3 +10x 2 -x-2 by x+2.
- The divisor x+2 has a root of -2
- Write the coefficients of the dividend: 5, 10, -1, -2.
- Perform synthetic division:
**Thus 5x 3 +10x 2 -x-2 divided by x+2 gives the result as 5x 2 -0.x+0 with a remainder of 0.
**5. Divide 3x 4 **+ 5x 3 **- 2x 2 **+ 7x - 1 by x - 3.
- The divisor x-3 has a root of 3
- Write the coefficients of the dividend: 3, 5, -2, 7, -1.
- Perform synthetic division:
**Thus 3x 4 +5x 3 -2x 2 +7x-1 divided by x-3 gives the result as 3x 3 +14x 2 +40x+127 with a remainder of 380.
**6. Divide 6x4-5x3+4x-3 by x+1
- The divisor x+1 has a root of -1
- Write the coefficients of the dividend: 6, -5, 4, -3, 2.
- Perform synthetic division:
**Thus 6x 4 -5x 3 +4x-3 divided buy x+1 gives the result as 6x 3 -11x 2 +15x-18 with a remainder of 20.
Synthetic Division of Polynomials practice Problems : Unsolved
Test your understanding of the concept by solving these Synthetic Division of Polynomials practice Problems
**Q1: Find Q(x) and R for the polynomial, P(x)= x3 - 6x + 8 divided by the linear factor x - 2.
**Q2: Find the quotient and remainder of the polynomial 8x2 - 4x + 7 when it is divided by x - 4.
**Q3: Solve the polynomial equation 2x5 + 5x4 - 4x3 + x2 -7x when it is divided by x-2.
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