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Synthetic Division Calculator

Synthetic Division Calculator

Dividend

Divisor (ax + b)

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An online synthetic division calculator will allow you to determine the reminder and quotient of polynomials using the synthetic division method. It also finds the zeros of the denominator and coefficient of the numerator.

Do you want to learn how to apply synthetic division on polynomials? Here we’ll teach you everything about the division of polynomial using synthetic division.

What is Synthetic Division of Polynomials?

Synthetic division is a simplified way of dividing polynomial with another polynomial expression of degree one and is commonly used for determining the zeros of the polynomial.

This technique is performed with less effort than the calculation of the long division method. A binomial equation is usually used as a divisor in the synthetic division method.

How to Perform Synthetic Division Method?

If you want to divide the polynomials using the synthetic method, you must be dividing it by a leading coefficient that should be a 1 or divide by a linear expression.

The requirements for the synthetic process method are:

  • The divisor of the given polynomial equation must have the degree of one.
  • The leading coefficient in the divisor should be also equal to one.

If the divisor of the leading coefficient is other than one, then the synthetic division will not be working well.
The basic technique to perform synthetic division is:

Bring down, multiply and add, multiply and add, multiply and add, ….

How to Divide Polynomials Using Synthetic Division?

You can do synthetic division manually but it’s a challenging task, however following steps are used by the synthetic division calculator for the synthetic process:

Step 1:

  • To find the number to substitute it in the division box, we need to set the denominator as zero.
  • If any term is missing, then write zero to fill in the missing term and write the numerator in descending order.

Step 2:

  • Bring the leading coefficient straight down when the problem is set up perfectly.

Step 3:

  • Now, substitute the outcomes in the next column by multiplying the number in the division box with the brought down number.

Step 4:

  • By substituting two numbers together, write the outcome at bottom of the row.

Step 5:

  • Write the final results.
  • The variables shall start with one power less than the denominator and go down with every term.

However, an online Quotient and Remainder Calculator will allow you to divide two numbers, a divided and a divisor to determine the quotient with a remainder.

Example:

Divide using synthetic division, when the dividend is 7x^3 + 4x + 8 and divisor (ax + b) is x + 2.

Solution:

\frac {7x^3 + 4x + 8} {x + 2}
Coefficient of the numerator polynomial

$$ 7, 4, 8 $$

The polynomial synthetic division calculator finds the zeros of denominator

$$ X + 2 = 0 $$

$$ X = −2.0 $$

Write down the problem in synthetic division format

\( \begin{array}{c|rrrrr}&x^{3}&x^{2}&x^{1}&x^{0}\\-2.0&7&0&4&8\\&&\\\hline&\end{array} \)

Carry down the leading coefficient to the bottom row

\( \begin{array}{c|rrrrr}-2.0&7&0&4&8\\&&\\\hline&7\end{array} \)

Now, synthetic substitution calculator multiplies the obtained value by the zero of the denominators, and put the outcome into the next column
$$ 7∗(−2.0) = −14 $$
\( \begin{array}{c|rrrrr}-2.0&7&0&4&8\\&&-14&\\\hline&7&\end{array} \)

Add down the column
$$ 0 + (−14) = −14 $$
\( \begin{array}{c|rrrrr}-2.0&7&0&4&8\\&&-14&\\\hline&7&-14&\end{array} \)

Multiply the obtained value by the zero of the denominators, and put the outcome into the next column
$$ −14 ∗ (−2.0) = 28 $$

\( \begin{array}{c|rrrrr}-2.0&7&0&4&8\\&&-14&28&\\\hline&7&-14&\end{array} \)
Add down the column
$$ 4 + (28) = 32 $$

\( \begin{array}{c|rrrrr}-2.0&7&0&4&8\\&&-14&28&\\\hline&7&-14&32&\end{array} \)
The synthetic division solver multiplies the obtained value by the zero of the denominators, and put the outcome into the next column
$$ 32 ∗ (−2.0 ) = −64 $$

\( \begin{array}{c|rrrrr}-2.0&7&0&4&8\\&&-14&28&-64&\\\hline&7&-14&32&\end{array} \)
Now, synthetic division calculator adds down the column
$$ 8 + (−64) = −56 $$

\( \begin{array}{c|rrrrr}-2.0&7&0&4&8\\&&-14&28&-64&\\\hline&7&-14&32&-56&\end{array} \)
So, the quotient is \( 7x^2−14x+32 \), and the remainder is −56

Therefore, Answer is:

$$ \frac{7x^3 + 4x + 8} {x + 2} $$
$$ 7x^2 − 14x + 32 − \frac {56} {x + 2} $$

However, an online LCM Calculator allows you to find the least common multiple (lcm) of a set of two, three, or more numbers.

Example:

Perform the synthetic division on polynomials, when the dividend is x^2 + 5x + 6 and divisor (ax + b) is x + 2.

Solution:

\frac { x^2 + 5x + 6} {x + 2}
Coefficient of the numerator polynomial
$$ 1, 5, 6 $$
Find the zeros of denominator
$$ X + 2 = 0 $$
$$ X = −2.0 $$
Write down the problem in synthetic division format

\( \begin{array}{c|rrrrr}& x^{2}&x^{1}&x^{0} \\-2.0& 1&5&6 \\&&\\\hline&\end{array} \)
Carry down the leading coefficient to the bottom row

\( \begin{array}{c|rrrrr}-2.0& 1&5&6 \\&&\\\hline&1\end{array} \)
The synthetic substitution calculator multiplies the obtained value by the zero of the denominators, and put the outcome into the next column

\( \begin{array}{c|rrrrr}-2.0&1&5&6\\&&-2&\\\hline&1&\end{array} \)
Now, polynomial synthetic division calculator adds down the column
$$ 5 + (-2) = 3 $$

\( \begin{array}{c|rrrrr}-2.0&1&5&6\\&&-2&\\\hline&1&3&\end{array} \)
The synthetic long division calculator multiplies the obtained value by the zero of the denominators, and put the outcome into the next column
$$ 3 ∗ (−2.0) = -6 $$

\( \begin{array}{c|rrrrr}-2.0&1&5&6\\&&-2&-6&\\\hline&1&3&\end{array} \)

Add down the column

$$ 6 + (-6) = 0 $$

\(  \begin{array}{c|rrrrr}-2.0&1&5&6\\&&-2&-6&\\\hline&1&3&0&\end{array} \)

So, the quotient is x + 3, and the remainder is 0

Therefore, Answer is:

$$ \frac{x^2 + 5x + 6} {x + 2} $$
$$ x + 3 + \frac {56} {x + 2} = x + 3 $$

How Synthetic Division Calculator Works?

An online synthetic substitution calculator divides the polynomial by binomial using synthetic division. Here we explain in steps how this calculator helps to determine the remainder and the quotient.

Input:

  • First, substitute the polynomials as dividend and divisor.
  • Click on the “Calculate” button.

Output:

  • The synthetic division polynomials calculator finds the coefficients of the numerator and the zero of the denominator.
  • It also provides the quotient and the remainder of polynomials.
  • The synthetic division of the polynomials calculator shows all steps of division using synthetic division.

FAQs:

Why synthetic division is important?

The synthetic division method plays a significant role for the division of polynomials in an effective and easy way as it breaks down the complex equations into simple equations.

What is the use of synthetic method?

The synthetic method is generally used for determining the zeros of the roots of the polynomials.

Can u always use synthetic method?

If the degree of the denominator is not equal to 1, then you cannot use the synthetic method. On the other side, if the denominator degree is greater than 1, then you should use long polynomial division.

What are the types of Polynomial Division?

There are four different types of Polynomial Division:

  • Polynomial Division by monomial
  • Polynomial Division by binomial
  • Polynomial Division by another polynomial
  • Monomial Division by another monomial

Conclusion:

Use an online synthetic division calculator to divide two different polynomials by binomial to find the remainder and the quotient of the division. Synthetic division is a shortcut way that divides the polynomials for the special case of dividing by the linear factor whose coefficient is one.

Reference:

Form the source of Wikipedia: Regular synthetic division, evaluating polynomials by the remainder theorem, Expanded synthetic division, For non-monic divisors, Compact Expanded Synthetic Division.

From the source of Lumen Learning: Two Polynomials, Use Synthetic Division to Divide, Divide A Second-Degree Polynomial, Divide A Third-Degree Polynomial, Using Synthetic Division to Divide a Fourth-Degree Polynomial.

From the source of Purple Math: Synthetic Division of Polynomials, Perform a Synthetic Division, Steps for Polynomial Synthetic Division Method, Advantages and Disadvantages of Synthetic Division Method.