Synthetic Division: Concept, Polynomials, Steps & Examples

Synthetic Division is one of the ways to perform Euclidean division of polynomials. Polynomials can be divided by two methods, viz: long division method and synthetic division method. 

• Synthetic division method is a shorter form of dividing the polynomials.
• The long division method is a longer form of dividing the polynomials.
• A polynomial is an expression made of variables, constants and exponents.
• It requires fewer calculations and less time to solve the problem. 
• It helps in finding zeros or the root of the polynomial.
• The synthetic division method is the shorter form of dividing polynomials when dividing by a linear factor.
• It is used to fix algorithms in CD players and telephones.
• It involves adding the required polynomial expression.

Read More Polynomials Formulas

Key Terms: Synthetic Division, Euclidean division, Polynomials, Long division method, Linear factor, Whole number, Coefficient, Degree

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Synthetic Division of Polynomials 

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Synthetic division method is a method that involves divison of polynomials with another polynomial equation with degree one. The method is used to count the number of zeroes of polynomials. 

• It involves division of polynomial p(x) by a linear factor (x - a), which is a polynomial of degree 1.
• Q(x) is the required quotient polynomial with a remainder R.
• Mathematically, Synthetic division method can be represented as:

p(x)/q(x) = p(x)/(x- a) = Quotient + (Remainder/(x - a))

p(x)/(x - a) = Q(x) + (R/(x - a))

Read More: Quadrant

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What are Polynomials

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Polynomial is a type of expression in mathematics in which exponents of all variables are a whole number. The degree of the polynomial is calculated by determining the highest value of the exponent.

• It involves various operations like addition, subtraction, multiplication and division.
• The power of any variable has to be a non-negative integer.

Read More: Rationalize the Denominator

The video below explains this:

Polynomials Detailed Video Explanation:

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What is the Division of Polynomials?

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Division in Polynomials is an arithmetic operation, where a higher degree polynomial is divided by a lower degree or equivalent degree polynomial. It may result as a polynomial or as a constant in answer. It is usually done to find out the roots or zeros of the Polynomials

Polynomial Division

Read More: Nature of Roots

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What Is Linear Factor?

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Linear Factor of a polynomial is a first degree polynomial which is the building block of higher degree polynomials. It appear in the form of ax+b. A linear factor of a polynomial is univariate meaning having one variable affecting the function.

Read More: Polynomials Formula

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How To Perform Synthetic Division

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Synthetic Division method is a shorter form of dividing polynomials. It is generally used to find out zeros or roots of the polynomials. This division method is performed with less effort.

• For synthetic Division method, the divisor of the polynomial should be degree 1 meaning it should be a linear factor.
• The coefficient of the divisor variable should also be equal to 1.
• Divide using the remainder theorem in conjunction with synthetic division.
• Set up the coefficient and then divide.

Read More: Eccentricity

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Steps of Synthetic Division

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The steps included in solving the synthetic division of polynomials are as follows:

Step 1

The setup is usually done as we divide in non-algebraic form. The divisor goes outside the box(divisor means the polynomial with which we are dividing) and the dividend goes inside the box.

• Dividends are always written in descending order and insert zeros for the missing terms.
• For example: (2x² – 3x – 1) divided by (x+1) starts with degree four and degree three and degree two is missing so we put zeros on that place.
• The said equation will be written as:

• Setting this said equation up as a synthetic equation:

 

Step 2

Bring down the first coefficient of the dividend.

Step 3

Multiply the coefficient of divisor with the brought down first coefficient of dividend then put the value just below the next coefficient.

Step 4

Add the column created.

Step 5

Repeat the steps that are multiplying the divisors coefficient with the previous dividends coefficient then add the same with the next coefficient.

Step 6

The numbers in the last row are coefficients of the quotient as well as the remainder. The final value on the right is the remainder.

• Working right to left, the next number is your constant.
• The next is the coefficient for x, the next is the coefficient for x squared and so on.
• The degree of the quotient is one less than the degree of the dividend.
• For example, if the degree of the dividend is 4, then the degree of the quotient is 3.

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Advantages of Synthetic Division

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• Synthetic Division is a form of dividing polynomials. 
• It is generally used to find out the zeros of the polynomial.
• Linear factor of a polynomial is a one degree polynomial.
• In synthetic division method, coefficient of the polynomials plays the most important part.
• When calculating synthetic division the next degree coefficient is absent then we put zeroes at that place.
• In the final equation, answer is always one degree less than the dividend.

Read More: Types of Functions

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Things to Remember

• Synthetic division method is a shorter form that involves the division of polynomials with another polynomial equation with degree one.
• It involves the division of polynomials with the linear factor.
• It is an error-prone method that can be solved without variables.
• Linear Factor is in the form of ax+b.
• The method is used to manually solve the Euclidean division of polynomials. 

Read More: Types of Relations

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