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Polynomial Synthetic Division

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Polynomial Synthetic Division


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Use synthetic division to determine the quotient and remainder from a given polynomial dividend and linear polynomial divisor.

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Now You Know

After completing this tutorial, you will be able to complete the following:

  • Divide a polynomial P(x) by x + b using synthetic division.
  • Divide a polynomial P(x) by ax + b where a is not 1 using synthetic division.

Everything You'll Have Covered

This Activity Object is designed to be an introduction to the operational procedure of polynomial synthetic division, while the guided practice in Sections 2 and 4 (with the potential for unlimited repetition) provide an effective strategy to help students gain mastery of this process, which will later be used to find the roots of - (cubic) and -degree polynomials.

Synthetic division is a method for finding real zeros of - (cubic) and -degree polynomial functions.

Synthetic division, in conjunction with the Rational Root Theorem, is perhaps the most important tool available to find real zeros of third- (cubic) and fourth-degree polynomial functions, which is one of the most difficult sections in the Algebra 2 curriculum. For example, in the cubic function , the possible zeros (according to the Rational Zero Theorem) are and using 1, in the synthetic division algorithm, produces the following quadratic function:

Because the remainder is zero, x = 1 is a root (or zero) of the cubic function with the other zeros being:

Therefore, the graph of passes through the points .

In this Activity Object, the process is initially explained (Section 1) by using the students' previous knowledge of polynomial long division, pointing out that, if the degree of the dividend is n and the degree of the divisor is m, then the degree of the quotient, using polynomial long division, is ; whereas in synthetic division, since the degree of the divisor is always 1 (i.e., the divisor is always a linear polynomial), the degree of the quotient is always . This last relationship, with respect to synthetic division, conceptually allows students to make the right choices (Sections 2 and 4) when asked to algebraically produce the correct quotient function at the end of the guided practice in Sections 2 and 4.

For example, given the 4th-degree polynomial

The degree of the quotient polynomial Q(x) is 4-1, or a third-degree polynomial. In addition to explaining the exponential relationship between the dividend, divisor, and quotient, Section 1 also describes the link between the degree of the divisor and remainder, which is one less than that of the divisor for polynomial long division and a constant for synthetic division.

A seven-step process can be used for polynomial synthetic division.

Section 2 of the Activity Object is where students are guided through the process of synthetic division. The following is a synopsis of this process for the dividend and divisor x-1:

Place coefficients of the dividend into the upper row (above the top line) of the synthetic division template. Remember: if a term is missing (i.e., in this example), use a zero.

Solve for x in the linear divisor and place the result to the left of the template.

Bring the leading coefficient ( ) down below the bottom line.

Multiply 1 and 5 together (the value of x, from the divisor times the leading coefficient).

Place the product from step 4 in the 2nd row, 2nd column (below -2, which is the coefficient of in the dividend).

Add the numbers in column 2 and vertically align the sum below the bottom line.

Repeat the process: multiply each remaining coefficient from the dividend by the value of x from the divisor, and use the results to write the resulting quotient polynomial:

Divide each term of the quotient polynomial by the coefficient of x in the divisor.

In the guided practice problems from Section 4, remember to divide each term of the quotient polynomial by the coefficient of x in the divisor, or else you will get an incorrect result.

For example, given the dividend and the divisor , find the quotient Q(x) and the remainder R(x).

It is important to emphasize that, because the final step of the guided practice in Section 4 may require the use of fractional coefficients, special care should be taken to monitor students' work to reduce potential mistakes to help them gain mastery with these types of problems, i.e.,

where D(x) is a linear polynomial divisor of the form ax+b, with

The following key vocabulary terms will be used throughout this Activity Object:

  • coefficient - A numerical or constant multiplier of the variable in an algebraic term, e.g., in the polynomial function , the number 2 is the coefficient of x
  • dividend - a polynomial that is to be divided by another polynomial of equal or lesser degree; for example, in the rational expression is the dividend
  • divisor - a polynomial that is divided into another polynomial of equal or greater degree; in the rational expression is the divisor
  • leading coefficient - the coefficient of a polynomial's leading term, e.g., 5 is the leading coefficient of the 4th-degree polynomial function
  • polynomial long division - in algebra, polynomial long division is an algorithm or operational procedure for dividing , where the degree of the dividend (P(x)) is greater than the degree of the divisor (D(x))
  • quotient - in polynomial long or synthetic division, the polynomial that results from the division of one polynomial by another polynomial of equal or lesser degree; the degree of the quotient is n-m, where n is the degree of the dividend and m is the degree of the divisor
  • remainder - in polynomial or synthetic long division, the part that is left over, e.g., when is divided by , the quotient is with a remainder of 5
  • synthetic division - in algebra, synthetic division is an algorithm or operational procedure for dividing , where the degree of the divisor (D(x)) is a linear polynomial.

Tutorial Details

Approximate Time 30 Minutes
Pre-requisite Concepts subtracting polynomials, multiplying polynomials, polynomial long division
Course Algebra-2
Type of Tutorial Procedural Development
Key Vocabulary polynomial function, linear polynomial divisor, polynomial synthetic division