Polynomial synthetic division

Synthetic division is a simplified and fast method for dividing a polynomial by a linear divisor of the type  x−c. It eliminates the requirement for lengthy division, making the operation faster and simpler, particularly for higher-degree polynomials. Here’s an explanation of how Polynomial synthetic division works, followed by an illustration.

Steps for Synthetic Division

Given a polynomial

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Set up the coefficients: Record the coefficients of the polynomial P(x) in a row. If any powers of  x are absent (e.g., no 𝑥 2  phrase), include a zero for that term.

Write the root of the divisor: The divisor $x-c$ means the root is $c$. This is the value you’ll use for the division.

Perform synthetic division:

• Bring down the first coefficient as it is.
• Multiply the result by 𝑐 and add it to the following coefficient.
• Repeat the method for each coefficient.

Interpret the results:

• The last number in the row is the remainder.
• The other numbers are the quotient’s coefficients.

Polynomial synthetic division examples: Divide

Step-by-step process

1. Write the coefficients: The polynomial 2x^3 – 3x^2 + 4x – 5 has the coefficients $2,-3,4,-5$.

2. Set up the synthetic division table:To divide by x−1, use 𝑐 = 1 .

3. Begin the synthetic division:

• Bring down the first coefficient (2).
• Multiply 2 by 1 (the root of the divisor x – 1) to obtain 2.
• Add this to the following coefficient:− 3 + 2 = − 1
• Multiplying -1 by 1 yields -1.
• Add this to the following coefficient: 4 + ( − 1 ) = 3 .
• Multiply 3 by 1 to get three.
• Add this to the final coefficient: −5 + 3 = −2

The synthetic division table now looks like this:

4. Interpret the results:

The quotient is  2x^ 2 -x+3.
The remaining is -2.

Thus, the division gives:

Important Points:
Quotient: The quotient’s coefficients are  2x^2 −x+3.
remaining:  -2 is the remaining.
Interpretation: This indicates that P(x)=(x−1)(2x^ 2 −x+3)−2, indicating that the division was correct with the exception of −2.

Why It Works:

Synthetic division uses the remainder theorem and polynomial properties to divide by a linear term ( x – c). Rather than going through the long division procedure step by step, synthetic division integrates all of the processes into a more compact form.

A complex number is one that takes the concept of numbers beyond the real number line. It consists of two parts: actual and imaginary.Four main features of complex arithmetic—modulus, argument, addition, and multiplication—will be discussed in this blog. It is often written in the following format:

1. The modulus of a complex number: The modulus of a complex number ∣z∣ indicates its distance from the origin (0, 0) in the complex plane. The modulus of a complex number (𝑧=𝑎+𝑏) is determined mathematically as follows:

Example: For the complex number z=3+4i, the modulus is calculated as:

This indicates that there are 5 units separating the origin from the point (3,4) in the complex plane.

2. Argument of a complex number:

The argument of a complex number is the angle $\theta$ that the complex number makes with the positive real axis in the complex plane. For $z=a+bi$, the argument θ is defined as:

Quadrant Considerations:

• Quadrant I (both a and b positive):

• Quadrant II ( a negative, b positive):

• Quadrant III (both a and b negative):

• Quadrant IV ( a positive, b negative):

Addition of complex number:

The addition of complex numbers is simple: add their real and imaginary parts.

If two complex numbers are given as:

Then their sum is

Steps for Addition

• Add the real parts: a+c
• Add the imaginary parts: b+d
• Combine the results into the form : (a+c)+(b+d)i

Example:

Mutiplication of two complex numbers:

Two complex numbers multiply using the distributive property and simplify depending on the knowledge that i^ 2 = −1.

If two complex numbers are

Then their product is:

By expanding the product:

Since i^2 = 1, the expression simplifies to:

Example:

Simplify i^2=-1:

Understanding the modulus and argument provides a geometric perspective of complex arithmetic, while addition and multiplication reveal their algebraic properties.