Synthetic Division of Polynomials

The Organic Chemistry Tutor
13 Feb 201810:32
EducationalLearning
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TLDRThis instructional video teaches viewers how to divide polynomial functions using synthetic division. It begins with a step-by-step example dividing a cubic polynomial by a linear factor, emphasizing the importance of zero coefficients and the correct order of terms. The video continues with additional examples, demonstrating the process for different polynomial degrees and highlighting the significance of the remainder in determining factorability. It concludes with a final example, reinforcing the method and emphasizing the importance of following the correct procedure for accurate results.

Takeaways
  • 📚 Synthetic division is a method for dividing polynomial functions.
  • 🔍 To begin synthetic division, identify the coefficients of the polynomial and the value that makes the divisor equal to zero.
  • 📝 Start the synthetic division process by placing the coefficients in descending order and the zero value to the left.
  • 🔢 Perform the synthetic division by multiplying, adding, and continuing the process until a remainder is obtained.
  • ✅ A zero remainder indicates that the divisor is a factor of the polynomial, meaning the polynomial is factorable by the divisor.
  • 📉 If the remainder is not zero, the polynomial is not factorable by the divisor, but division can still be performed.
  • 📈 After division, the quotient is the resulting polynomial, and if there's a remainder, it's expressed as 'remainder divided by the divisor'.
  • 📋 It's crucial to remember to include zero coefficients for missing powers of x when setting up the synthetic division.
  • 📉 In the case of a non-zero remainder, the division result includes the quotient and the remainder term.
  • 📝 The script provides step-by-step examples, demonstrating the synthetic division process with different polynomials and divisors.
  • 🎓 The video aims to teach viewers how to divide polynomial functions using synthetic division and provides practice examples for understanding.
Q & A

  • What is the main topic of the video?

    -The main topic of the video is dividing polynomial functions using synthetic division.

  • What is the first polynomial function given in the video?

    -The first polynomial function given is x^3 - 2x^2 - 5x + 6.

  • How is synthetic division initiated in the video?

    -Synthetic division is initiated by identifying a zero of the function, which is found by setting the divisor equal to zero and solving for x.

  • What is the divisor used in the first example of synthetic division in the video?

    -The divisor used in the first example is x - 3.

  • What does a remainder of zero in synthetic division indicate?

    -A remainder of zero indicates that the divisor is a factor of the polynomial function, meaning the function is factorable by that divisor.

  • What is the result of dividing the first polynomial function by x - 3 using synthetic division?

    -The result of dividing x^3 - 2x^2 - 5x + 6 by x - 3 is x^2 + x - 2.

  • What is the second polynomial function discussed in the video?

    -The second polynomial function discussed is x^3 + 5x^2 + 7x + 2.

  • What is the divisor used in the second example of synthetic division in the video?

    -The divisor used in the second example is x + 2.

  • What does the process of synthetic division involve?

    -The process of synthetic division involves bringing down the leading coefficient, multiplying and adding sequentially, and repeating this process until the division is complete.

  • What happens when the remainder is not zero after synthetic division?

    -When the remainder is not zero, it indicates that the divisor is not a factor of the polynomial. The division process still yields a quotient and a remainder term, which is the remainder divided by the divisor.

  • How is the remainder handled in the division process when it is not zero?

    -The remainder is added to the quotient as a separate term, which is the remainder divided by the divisor, when the remainder is not zero.

  • What is the importance of writing the polynomial function in descending order when performing synthetic division?

    -Writing the polynomial function in descending order ensures that the synthetic division process is performed correctly, especially when there are missing terms like x^2 or x in the polynomial.

  • What is the final answer for the division of 3x^4 - 5x^2 + 6 by x - 2 using synthetic division?

    -The final answer for the division is 3x^3 + 6x^2 + 7x + 14, with a remainder of 34, which is then expressed as 34 divided by x - 2.

Outlines
00:00
📚 Introduction to Synthetic Division

The video begins by introducing the concept of synthetic division, a method used to divide polynomial functions. The first example provided is dividing a cubic polynomial (x^3 - 2x^2 - 5x + 6) by a linear factor (x - 3). The process involves setting up coefficients in a synthetic division table and using the zero of the divisor (x = 3) to perform the division. The video demonstrates the 'multiply, add, multiply, add' sequence, revealing that the remainder is zero, indicating that x - 3 is a factor of the polynomial. The result of the division is a quadratic polynomial (x^2 + x - 2).

05:02
🔍 Dividing Polynomials Using Synthetic Division

The second paragraph continues the theme of synthetic division by presenting additional examples. It includes dividing a cubic polynomial (x^3 + 5x^2 + 7x + 2) by a linear factor (x + 2) and a quadratic polynomial (3x^2 + 7x - 20) by another linear factor (x + 5). The video illustrates the synthetic division process, emphasizing the importance of setting the divisor to zero to find the zero of the function and correctly placing coefficients in descending order. It also explains how to handle non-zero remainders by expressing the result as a polynomial plus a remainder divided by the divisor.

10:03
📘 Advanced Synthetic Division Examples

This paragraph presents more complex examples of synthetic division, including dividing a cubic polynomial (7x^3 + 6x - 8) by a linear factor (x - 4) and a quartic polynomial (3x^4 - 5x^2 + 6) by a linear factor (x - 2). The video reminds viewers to include the zero x^2 term when setting up the division for cubic polynomials and to be mindful of the order of coefficients. It demonstrates the synthetic division process, including calculations for each step, and concludes with the final answers for both examples, showing the quotient polynomials and remainders.

👋 Conclusion and Thanks for Watching

The final paragraph wraps up the video by summarizing the key points covered in the tutorial on synthetic division. It reiterates the importance of the method for dividing polynomial functions and thanks the viewers for watching. The video concludes with an encouragement for viewers to practice the technique and apply it to their own problems.

Mindmap
Keywords
💡Synthetic Division
Synthetic division is a simplified method of polynomial division used to determine if a polynomial can be divided by a linear factor of the form x - c. In the video, synthetic division is the main technique discussed for dividing polynomial functions. It is illustrated through various examples, such as dividing x^3 - 2x^2 - 5x + 6 by x - 3, where the process involves placing the coefficients of the polynomial and the root of the divisor (in this case, 3) into a systematic layout to perform the division.
💡Polynomial Functions
A polynomial function is a mathematical expression involving a sum of powers in a single variable, with coefficients as constants. The video script focuses on dividing these functions using synthetic division. For example, the polynomial x^3 - 2x^2 - 5x + 6 is a cubic polynomial, and the process of dividing it by a linear factor is explained in the video.
💡Coefficients
In the context of polynomial functions, coefficients are the numerical factors that multiply the variables raised to various powers. The script describes how to use the coefficients of a polynomial in synthetic division. For instance, when dividing x^3 - 2x^2 - 5x + 6 by x - 3, the coefficients 1, -2, -5, and 6 are used in the synthetic division process.
💡Zeros
A zero of a function is a value of x for which the function equals zero. In the script, finding zeros is crucial for synthetic division because it helps identify potential factors of the polynomial. For example, when dividing by x - 3, the zero is x = 3, which is used to set up the synthetic division.
💡Factorable
A polynomial is said to be factorable by a certain expression if it can be divided by that expression without leaving a remainder. The script explains that if the remainder is zero after synthetic division, the divisor is a factor of the polynomial. For instance, x^3 + 5x^2 + 7x + 2 is factorable by x + 2 because the remainder is zero after performing synthetic division.
💡Remainder
In division, the remainder is what is left over after the division process if the dividend cannot be evenly divided by the divisor. In the context of synthetic division, if the final result is not zero, it indicates that the polynomial is not factorable by the given linear expression. For example, when dividing 3x^2 + 7x - 20 by x + 5, the remainder is 20, indicating that x + 5 is not a factor.
💡Divisor
The divisor is the expression by which another expression (the dividend) is divided. In synthetic division, the divisor is typically a linear factor like x - c. The script demonstrates how to use the divisor in the synthetic division process, such as dividing polynomials by x - 3, x + 2, and x + 5.
💡Quadratic
A quadratic expression is a polynomial of degree two. In the video, after dividing a cubic polynomial by a linear factor, the result is often a quadratic expression, such as x^2 + x - 2, which is the result of dividing x^3 - 2x^2 - 5x + 6 by x - 3.
💡Descendant Order
When writing polynomials for synthetic division, it is important to arrange the coefficients in descendant order, which means from the highest power of x to the constant term. The script emphasizes the importance of including all terms, including those with zero coefficients, to ensure the correct result, as seen when dividing 7x^3 + 6x - 8 by x - 4, where a zero x^2 term is included.
💡Root
A root of a polynomial is a value for which the polynomial equals zero. In the context of synthetic division, finding the root of the divisor helps in setting up the division process. For example, the script mentions finding the root of x - 3, which is 3, to perform the division of the polynomial x^3 - 2x^2 - 5x + 6.
Highlights

Introduction to dividing polynomial functions using synthetic division.

Example provided: dividing x^3 - 2x^2 - 5x + 6 by x - 3.

Explanation of placing coefficients and the zero of the divisor in synthetic division.

Step-by-step guide through the synthetic division process.

Verification of factorability by checking the remainder after synthetic division.

Result interpretation: if remainder is zero, the divisor is a factor.

Demonstration of dividing x^3 + 5x^2 + 7x + 2 by x + 2.

Practice prompt for viewers to try synthetic division themselves.

Division of 3x^2 + 7x - 20 by x + 5 with a non-zero remainder.

Explanation of handling non-zero remainders in synthetic division.

Division of 7x^3 + 6x - 8 by x - 4 with detailed synthetic division steps.

Emphasis on writing polynomials in descending order for accuracy.

Division of 3x^4 - 5x^2 + 6 by x - 2, including handling of zero terms.

Final answer presentation for each synthetic division example.

Encouragement to practice and the conclusion of the tutorial.

Transcripts

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