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.

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 2x³ + 3x² + 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, 2x³+3x²+5x+6 divided by x-2 gives 2x²+7x+9 with a remainder of 24 as the final answer.

2. Divide 4x^3-6x²+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 4x³-6x²+x-8 divided by x+1 gives the result as 4x²-10x+11 with -19 as a remainder.

3. Divide -3x³+4x²-2x+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 -3x³+4x²-2x+5 divided by x-1 gives x³-2x²+2x+0 with a remainder of 5.

4. Divide 5x³+10x²-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 5x³+10x²-x-2 divided by x+2 gives the result as 5x²-0.x+0 with a remainder of 0.

5. Divide 3x⁴ + 5x³ – 2x² + 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 3x⁴+5x³-2x²+7x-1 divided by x-3 gives the result as 3x³+14x²+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 6x⁴-5x³+4x-3 divided buy x+1 gives the result as 6x³-11x²+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.

Read More,

• Arithmetic
• Arithmetic Operations
• Polynomial
• Long Division of Polynomial

Synthetic Division of Polynomials Practice Problems- FAQs

What do you mean by synthetic division of Polynomials?

Synthetic division is a simplified method of dividing polynomials, particularly useful for dividing the polynomial by a linear factor of the form x−c , where c is a constant.

What are the applications of synthetic division method?

Synthetic division is mainly used to find the zeroes of roots of polynomials.

When do you use synthetic division method?

Synthetic division is used when a polynomial is to be divided by a linear expression and the leading coefficient (first number) must be a 1. For example, any polynomial equation of any degree can be divided by x + 1 but not by x²+1

What do we do if a polynomial is missing a term?

If a polynomial is missing a term the you must include a 0 for the missing coefficient when setting up synthetic division. For example you can write x² – 7 as x² + 0x – 7.

Can we use synthetic division for complex or irrational roots?

Synthetic division is primarily used for real, rational roots. For complex or irrational roots, other methods such as polynomial long division or numerical algorithms are more appropriate.

How do We interpret the result when using synthetic division method?

The numbers in the final row (excluding the last number) are the coefficients of the quotient polynomial. The last number in the row is the remainder.