In mathematics, a synthetic division is a well-known technique in algebraic equations to do polynomial division in a simplified way. It is a technique to divide the polynomials with the help of the coefficients of the polynomials.
In algebra, this process is used to simplify the process of polynomial division with the system of coefficients. Polynomials are algebraic equations in which the variables are raised to the power and take its product by coefficients.
In this article, we’ll learn and explain the term synthetic division with the help of examples.
In algebra, the synthetic division is a method for dividing polynomials with the help of its factors. This algebraic equation technique is also known as the short division method. The main purpose of synthetic division is to simplify the process of polynomial long division.
It is the shortened method to find the quotient and remainder of the division of polynomials as the polynomial division is used frequently in algebra for various purposes. When the linear factors are available then this technique is used to get the results in a few calculations by dividing the coefficients of a polynomial by the linear factor.
The formula of synthetic division is:
| P(x)/(x-a) = Q(x) + R/(x-a) | Where P(x) is the given polynomial equationx – a is the linear factorQ(x) is the quotient of the polynomial division R is the remainder of the polynomial division |
In mathematics, the synthetic division is a faster way to divide the polynomial as compared to the polynomial long division method. Here are a few steps to perform synthetic division.
First step
First of all, arrange the terms of the given polynomials from least to greatest order. Keep one thing in mind, if there is any missing term while writing a term from larger to smaller write the missing power of x in the polynomial filled with zeros. Such as
A polynomial expression 3x4 – 3x2 + 2x3 + 1 divided by x – 2 is given then write it in descending order and write the missing term.
3x4 – 3x2 + 2x3 + 1 will be written as 3x4 + 2x3 – 3x2 + 0x + 1
Dividend = 3x4 + 2x3 – 3x2 + 0x + 1
Divisor = x – 2
Write the constant coefficient of the polynomial expression inside the division symbol.
| 32 -3 0 1 | |
Second step
Now write the divisor outside the symbol of division. Such as the given divisor is x – 2 then you have to take this divisor equal to zero and find the value of x that will be written outside the notation of division. Such as x – 2 = 0 → x = 2
| 2 | 32 -3 0 1 |
Third step
Multiply the divisor “2” by the leading coefficient of the given polynomial “3”, and write the result under the next coefficient in the polynomial.
| 2 | 32 -3 0 1 ↓ 6 |
Fourth step
Add the multiplied result to the 2nd coefficient of the expression and write the sum below it while writing the leading coefficient as it is.
| 2 | 32 -3 0 1 ↓ 6 |
| 3 8 |
Fifth step
Continue to multiply the term and add it to the next term until you have covered all the coefficients. In other words, repeat the step third and fourth until the last coefficient.
| 2 | 3 2 -3 0 1 ↓ 6 16 26 52 |
| 3 8 13 26| 53 |
Sixth step
Write the final result, with the remainder on top and the coefficients of the quotient below.
Q(x) = 3x3 + 8x2 + 13x + 26
R = 53
Hence
Synthetic DivisionBenefits
Synthetic division is a well-known method in algebraic expressions that has many benefits over other methods of dividing polynomials.
| Speed and accuracy of synthetic division | Speed and accuracy of long division |
| Synthetic division, involves only simple addition and multiplication, making it a faster and more accurate method for dividing polynomials. | Long division involves a lot of manual computation, including multiplying and subtracting polynomials, which can be time-consuming and prone to errors. |
| Efficient method for finding factors and roots of polynomials | |
| Synthetic division is also an efficient method for finding factors and roots of polynomials. By performing synthetic division on a polynomial with a linear factor, we can quickly determine whether the polynomial has a root at that factor. If the remainder of the division is zero, then the linear factor is a root of the polynomial. This property is particularly useful in solving polynomial equations, as it can help us narrow down the potential roots of the equation. | |
| Applications in real-world problems | |
| In engineering, mathematics, & physics Synthetic division has several applications in real-world problems. For instance, it can be used to analyze the stability of control systems by calculating the characteristic equation of the system. Synthetic division can also be used to approximate the behavior of a system under different conditions by generating transfer functions. |
Examples of synthetic division
Here are a few examples of synthetic division.
Example 1
Divide the polynomial P(x) = 5x3 + 3x4 + 4x5 – 2x + 6x2 + 12 by (x – 3).
Solution
Step 1: Arrange the polynomial in descending order and write their coefficients.
P(x) = 4x5 + 3x4 + 5x3 + 6x2– 2x + 12
Coefficients of polynomial = 4, 3, 5, 6, -2, 12
| 4 3 5 6 -2 12 | |
Step 2:Now take the linear factor and find the value of the unknown.
x – 3 = 0
x = 3
| 3 | 4 3 5 6 -2 12 |
Step 3:Multiply the divisor “3” by the leading coefficient of the given polynomial “4”, and write the result under the next coefficient in the polynomial.
| 3 | 4 3 5 6 -2 12 12 |
Step 4:Add the multiplied result to the 2nd coefficient of the expression and write the sum below it while writing the leading coefficient as it is.
| 3 | 4 3 5 6 -2 12 ↓ 12 |
| 4 15 |
Step 5:Now repeat steps 3 & 4 to complete the division.
| 3 | 4 3 5 6 -2 12 ↓ 12 36 123 387 1155 |
| 4 15 41 129 385 | 1167 |
Step 6: Write the final result, with the remainder on top and the coefficients of the quotient below.
Q(x) = 4x4 + 15x3 + 41x2 + 129x + 385
R = 1167
Hence,
P(x)/ (x – a) = Q(x) + R/ (x – a)
[5x3 + 3x4 + 4x5 – 2x + 6x2 + 12] / (x – 3) = [4x4 + 15x3 + 41x2 + 129x + 385] + 1167/(x – 3)
A synthetic division calculator is an alternative way to solve the problems of polynomial division without involving into lengthy calculations.
Example 2
Divide the polynomial P(y) = 2y2 – 4y3 – y – 11 by (y – 1).
Solution
Step 1: Arrange the polynomial in descending order and write their coefficients.
P(y) = – 4y3 + 2y2– y – 11
Coefficients of polynomial = -4, 2, -1, -11
| -4 2 -1 -11 | |
Step 2:Now take the linear factor and find the value of the unknown.
y – 1 = 0
y = 1
| 1 | -4 2 -1 -11 |
Step 3:Multiply the divisor “1” by the leading coefficient of the given polynomial “-4”, and write the result under the next coefficient in the polynomial.
| 1 | -4 2 -1 -11 -4 |
Step 4: Add the multiplied result to the 2nd coefficient of the expression and write the sum below it while writing the leading coefficient as it is.
| 1 | -4 2 -1 -11 ↓ -4 |
| -4 -2 |
Step 5: Now repeat steps 3 & 4 to complete the division.
| 1 | -4 2 -1 -11 ↓ -4 -2 -3 |
| -4 -2 -3 | -14 |
Step 6: Write the final result, with the remainder on top and the coefficients of the quotient below.
Q(x) = -4y2 – 2y – 3
R = -14
Hence,
P(x)/ (x – a) = Q(x) + R/ (x – a)
[-4y3 + 2y2 – y – 11] / (y – 1) = [-4y2 – 2y – 3] – 14/(y – 1)
Conclusion
Now you can take assistance from this post to learn the basics and calculations of the synthetic division to calculate the quotient and remainder of polynomial division. There is also some solved examples available above to understand the topic briefly.
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