Subjects

Algebra 2

Dividing polynomials & quadratics

Dividing quadratics by linear factors

Dividing quadratics by linear expressions with remainders: missing x-term

Step by Step Synthetic Division Tutorial: Easy Guide with Examples and Answers

Name: Knowunity
Brand: Knowunity

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Step by Step Synthetic Division Tutorial: Easy Guide with Examples and Answers

George Yassa

@georgeyassa_gqtt

3 Followers

Synthetic division is a step-by-step method for dividing polynomials by linear factors. This technique simplifies the process of polynomial long division, especially when dealing with monic linear divisors. The transcript provides a comprehensive overview of synthetic division, including its definition, application, and examples.

Key points:

Synthetic division is used for dividing polynomials by monic linear divisors (x - c)
It's a shortcut method that reduces the work compared to traditional long division
The process involves arranging coefficients and performing simple arithmetic operations
Synthetic division is particularly useful for finding polynomial roots and factoring

2/3/2023

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3.3B Synthetic Division

This section introduces the concept of synthetic division, a simplified method for dividing polynomials by linear factors. It begins by defining key terms and progresses to demonstrate the technique through examples.

A. Linear Polynomials

Linear polynomials are defined as polynomials of the form ax + b, where a ≠ 0.

Example: 3x - 1, 2x - 5, and 3x - 1 are linear polynomials.

B. Monic Polynomials

Monic polynomials are polynomials with a leading coefficient of 1.

Example: x³ + 5x² - 2x - 1, x² - 3x² - 2, and x + b are monic polynomials.

C. Monic Linear Polynomials

Monic linear polynomials are both linear and monic, taking the form x - c.

Example: x + 5, x - 2, and x + 3 are monic linear polynomials.

Highlight: All monic linear polynomials can be written as x - c, where c can be positive, negative, or zero.

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Synthetic Division

This page introduces the concept of synthetic division as a shortcut method for dividing polynomials by monic linear divisors.

E. Synthetic Division Process

The page demonstrates how to perform synthetic division using the same example from the previous page: 2x³ - 4x² + 7x - 11 ÷ (x + 2).

Definition: Synthetic division is a simplified method for dividing polynomials by monic linear divisors of the form x - c.

Example: Dividing 2x³ - 4x² + 7x - 11 by x + 2 using synthetic division: -2) 2 -4 7 -11 -4 16 -46 2 -8 23 -57 Answer: 2x² - 8x + 23 + 26/(x + 2)

Highlight: Synthetic division only works for monic linear divisors. For all other cases, algebraic long division must be used.

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Long Division Example with Monic Linear Divisor

This page demonstrates the process of long division for polynomials using a monic linear divisor, highlighting the steps involved and introducing the concept of synthetic division.

D. Long Division Example

The page shows a detailed example of dividing 2x³ - 4x² + 7x - 11 by x + 2 using traditional long division.

Example: 2x³ - 4x² + 7x - 11 ÷ (x + 2) = 2x² - 8x + 23 + 26/(x + 2)

The process is then simplified by focusing only on the coefficients and changing the divisor to "-2" to facilitate addition instead of subtraction.

Highlight: This simplification process is a step towards synthetic division, which further streamlines the calculation.

View

Examples of Synthetic Division

This page provides two examples of synthetic division to reinforce understanding of the technique.

F. Examples

Example 1: Divide 3x³ - x + 5 by x - 1 using synthetic division

Example:

3 0 -1 5 3 3 2 3 3 2 7 Answer: 3x² + 3x + 2 + 7/(x - 1)

Example 2: Divide 2x³ + 6x² + 5x - 3 by x + 3 using synthetic division

Example: -3) 2 6 5 -3 -6 0 -15 2 0 5 -18 Answer: 2x² + 5 - 18/(x + 3)

These examples demonstrate the efficiency of synthetic division compared to traditional long division methods for polynomials.

Highlight: Synthetic division is a powerful tool for quickly dividing polynomials by linear factors, making it invaluable for solving polynomial equations and finding roots.

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Knowunity is the # 1 ranked education app in five European countries

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Students use Knowunity

#1

In Education App Charts in 12 Countries

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I love this app so much [...] I recommend Knowunity to everyone!!! I went from a C to an A with it :D

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The application is very simple and well designed. So far I have found what I was looking for :D

SuSSan, iOS User

Love this App ❤️, I use it basically all the time whenever I'm studying

Subjects

Algebra 2

Dividing polynomials & quadratics

Dividing quadratics by linear factors

Dividing quadratics by linear expressions with remainders: missing x-term

Step by Step Synthetic Division Tutorial: Easy Guide with Examples and Answers

George Yassa

@georgeyassa_gqtt

3 Followers

Key points:

Synthetic division is used for dividing polynomials by monic linear divisors (x - c)
It's a shortcut method that reduces the work compared to traditional long division
The process involves arranging coefficients and performing simple arithmetic operations
Synthetic division is particularly useful for finding polynomial roots and factoring

2/3/2023

Algebra 2

450

3.3B Synthetic Division

A. Linear Polynomials

Linear polynomials are defined as polynomials of the form ax + b, where a ≠ 0.

Example: 3x - 1, 2x - 5, and 3x - 1 are linear polynomials.

B. Monic Polynomials

Monic polynomials are polynomials with a leading coefficient of 1.

Example: x³ + 5x² - 2x - 1, x² - 3x² - 2, and x + b are monic polynomials.

C. Monic Linear Polynomials

Monic linear polynomials are both linear and monic, taking the form x - c.

Example: x + 5, x - 2, and x + 3 are monic linear polynomials.

Highlight: All monic linear polynomials can be written as x - c, where c can be positive, negative, or zero.

Synthetic Division

This page introduces the concept of synthetic division as a shortcut method for dividing polynomials by monic linear divisors.

E. Synthetic Division Process

The page demonstrates how to perform synthetic division using the same example from the previous page: 2x³ - 4x² + 7x - 11 ÷ (x + 2).

Definition: Synthetic division is a simplified method for dividing polynomials by monic linear divisors of the form x - c.

Example: Dividing 2x³ - 4x² + 7x - 11 by x + 2 using synthetic division: -2) 2 -4 7 -11 -4 16 -46 2 -8 23 -57 Answer: 2x² - 8x + 23 + 26/(x + 2)

Highlight: Synthetic division only works for monic linear divisors. For all other cases, algebraic long division must be used.

Long Division Example with Monic Linear Divisor

This page demonstrates the process of long division for polynomials using a monic linear divisor, highlighting the steps involved and introducing the concept of synthetic division.

D. Long Division Example

The page shows a detailed example of dividing 2x³ - 4x² + 7x - 11 by x + 2 using traditional long division.

Example: 2x³ - 4x² + 7x - 11 ÷ (x + 2) = 2x² - 8x + 23 + 26/(x + 2)

The process is then simplified by focusing only on the coefficients and changing the divisor to "-2" to facilitate addition instead of subtraction.

Highlight: This simplification process is a step towards synthetic division, which further streamlines the calculation.

Examples of Synthetic Division

This page provides two examples of synthetic division to reinforce understanding of the technique.

F. Examples

Example 1: Divide 3x³ - x + 5 by x - 1 using synthetic division

Example:

3 0 -1 5 3 3 2 3 3 2 7 Answer: 3x² + 3x + 2 + 7/(x - 1)

Example 2: Divide 2x³ + 6x² + 5x - 3 by x + 3 using synthetic division

Example: -3) 2 6 5 -3 -6 0 -15 2 0 5 -18 Answer: 2x² + 5 - 18/(x + 3)

These examples demonstrate the efficiency of synthetic division compared to traditional long division methods for polynomials.

Highlight: Synthetic division is a powerful tool for quickly dividing polynomials by linear factors, making it invaluable for solving polynomial equations and finding roots.

Can't find what you're looking for? Explore other subjects.

Knowunity is the # 1 ranked education app in five European countries

Knowunity was a featured story by Apple and has consistently topped the app store charts within the education category in Germany, Italy, Poland, Switzerland and United Kingdom. Join Knowunity today and help millions of students around the world.

Download in

Google Play

Download in

App Store

4.9+

Average App Rating

13 M

Students use Knowunity

#1

In Education App Charts in 12 Countries

950 K+

Students uploaded study notes

Still not sure? Look at what your fellow peers are saying...

iOS User

I love this app so much [...] I recommend Knowunity to everyone!!! I went from a C to an A with it :D

Stefan S, iOS User

The application is very simple and well designed. So far I have found what I was looking for :D

SuSSan, iOS User

Step by Step Synthetic Division Tutorial: Easy Guide with Examples and Answers

3.3B Synthetic Division

A. Linear Polynomials

B. Monic Polynomials

C. Monic Linear Polynomials

Synthetic Division

E. Synthetic Division Process

Long Division Example with Monic Linear Divisor

D. Long Division Example

Examples of Synthetic Division

F. Examples

Example 1: Divide 3x³ - x + 5 by x - 1 using synthetic division

Example 2: Divide 2x³ + 6x² + 5x - 3 by x + 3 using synthetic division

Similar Content

Can't find what you're looking for? Explore other subjects.

Knowunity is the # 1 ranked education app in five European countries

Knowunity was a featured story by Apple and has consistently topped the app store charts within the education category in Germany, Italy, Poland, Switzerland and United Kingdom. Join Knowunity today and help millions of students around the world.

4.9+

13 M

#1

950 K+

Still not sure? Look at what your fellow peers are saying...

I love this app so much [...] I recommend Knowunity to everyone!!! I went from a C to an A with it :D

The application is very simple and well designed. So far I have found what I was looking for :D

Love this App ❤️, I use it basically all the time whenever I'm studying

Step by Step Synthetic Division Tutorial: Easy Guide with Examples and Answers

3.3B Synthetic Division

A. Linear Polynomials

B. Monic Polynomials

C. Monic Linear Polynomials

Synthetic Division

E. Synthetic Division Process

Long Division Example with Monic Linear Divisor

D. Long Division Example

Examples of Synthetic Division

F. Examples

Example 1: Divide 3x³ - x + 5 by x - 1 using synthetic division

Example 2: Divide 2x³ + 6x² + 5x - 3 by x + 3 using synthetic division

Similar Content

Can't find what you're looking for? Explore other subjects.

Knowunity is the # 1 ranked education app in five European countries

Knowunity was a featured story by Apple and has consistently topped the app store charts within the education category in Germany, Italy, Poland, Switzerland and United Kingdom. Join Knowunity today and help millions of students around the world.

4.9+

13 M

#1

950 K+

Still not sure? Look at what your fellow peers are saying...

I love this app so much [...] I recommend Knowunity to everyone!!! I went from a C to an A with it :D

The application is very simple and well designed. So far I have found what I was looking for :D

Love this App ❤️, I use it basically all the time whenever I'm studying