How to Divide Polynomials Using Synthetic Division?

When a polynomial has to be divided by a linear factor, the synthetic division is the shortest method. In this article, let's familiarize ourselves with dividing polynomials using synthetic division.

How to Divide Polynomials Using Synthetic Division?

In mathematics, there are two different ways to divide polynomials. One is the long division method. Another method is synthetic division. Between the two methods, the shortcut method for dividing polynomials is the synthetic division method.

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Step by step guide to dividing polynomials by using synthetic division

Synthetic division is a method used to perform the division operation on polynomials when the divisor is a linear factor. One of the advantages of using this method over the traditional long method is that the synthetic division allows the person to calculate variables without writing them while doing polynomial division, which also makes it easier than the long method.

Synthetic division of polynomials definition

When we divide a polynomial \(p(x)\) by a linear factor \((x – a)\) (which is a polynomial of degree \(1\)), \(Q(x)\) is the quotient polynomial and \(R\) is the remainder.

\(\color{blue}{\frac{p(x)}{q(x)} = \frac{p(x)}{(x- a)} = Quotient + (\frac{Remainder}{(x – a)})}\)

\(\color{blue}{\frac{p(x)}{(x – a)} = Q(x) + (\frac{R}{(x – a)})}\)

We use synthetic division to evaluate polynomials by the remainder theorem, wherein we evaluate the value of \(p(x)\) at \(a\) while dividing \((\frac{p(x)}{(x – a)})\). That is, to find if \(a\) is the factor of the polynomial \(p(x)\), use the synthetic division to find the remainder quickly.

Example: \((x^2+ x – 2) ÷ (x + 2)\)

Solution:

  • Step 1: Write the coefficients of the dividend inside the box and zero of \(x + 2\) as the divisor.
  • Step 2: Bring down the leading coefficient \(1\) to the bottom row.
  • Step 3: Multiply \(-2\) by \(1\) and write the product \(-2\) in the middle row.
  • Step 4: Add \(1\) and \(-2\) in the second column and write the sum \(-1\) in the bottom row.
  • Step 5: Now, multiply \(-2\) by \(-1\) (obtained in step \(4\)) and write the product \(2\) below \(-2\).
  • Step 6: Add \(-2\) and \(2\) in the third column and write the sum \(0\) in the bottom row.
  • Step 7: The bottom row gives the coefficient of the quotient. The degree of the quotient is one less than that of the dividend. So, the final answer is \(\frac{x-1+ 0}{(x + 2)}= x – 1\).

Please note that the last box in the bottom row shows the remainder.

Synthetic division vs long division

By comparing both methods, let’s see how long division differs from the synthetic division of polynomials. In the following example, we divide the polynomial \(4x^2-5x -21\) by a linear polynomial \(x- 3\).

In the example given below, another polynomial \(2x^2+ 3x -1\) is divided by a linear polynomial \(x +1\). When a polynomial \(P(x)\) is to be divided by a linear factor, we write the coefficients alone, bring down the first coefficient, multiply, and add. Repeat the multiplication and addition to get to the end of the polynomial.

Using synthetic division, we can do complex divisions and easily find solutions.

Synthetic division method

The following steps are taken when performing synthetic division and finding the quotient and remainder. For a better understanding, we consider the following expression as a reference: \(\frac{(2x^3-3x^2 + 4x+ 5)}{(x + 2)}\)

  • Check if the polynomial is in the standard form.
  • Write the coefficients in the dividend’s place and write the zero of the linear factor in the divisor’s place.
  • Bring the first coefficient down.
  • Multiply it with the divisor and write it below the next coefficient.
  • Add them and write the value below.
  • Repeat the previous \(2\) steps until you reach the last term.
  • Separate the last term thus obtained which is the remainder.
  • Now group the coefficients with the variables to get the quotient.

Therefore, the result obtained after synthetic division of \(\frac{(2x^3-3x^2 + 4x + 5)}{(x + 2)}\) is \(2x^2 -7x +18\) and remainder is \(-31\).

Note that the resultant polynomial is of one order less than the dividend polynomial.

Dividing Polynomials by Using Synthetic Division – Example 1:

Solve by using the synthetic division method. \(\frac{3x^3+5x-1}{x+1}\)

Exercises for Dividing Polynomials by Using Synthetic Division

Perform synthetic division to find the quotient of the following expression.

  1. \(\color{blue}{\frac{4x^3-8x^2-x+5}{2x-1}}\)
  2. \(\color{blue}{\frac{x^3-5x^2+3x+7}{x-3}}\)
  3. \(\color{blue}{\frac{8x^3+12x^2-2x}{4x^2-1}}\)
This image has an empty alt attribute; its file name is answers.png
  1. \(\color{blue}{2x^2-3x-2+\frac{3}{2x-1}}\)
  2. \(\color{blue}{x^2-2x-3-\frac{2}{x-3}}\)
  3. \(\color{blue}{2x+3+\frac{3}{4x^2-1}}\)

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