How To Use Synthetic Division on Any Polynomial!

The Organic Chemistry Tutor
7 Oct 201810:14
EducationalLearning
32 Likes 10 Comments

TLDRThis educational video tutorial demonstrates the process of synthetic division for polynomials. It begins with dividing a cubic polynomial by a linear one, showing the steps to achieve a quotient and remainder. The video continues with more examples, illustrating how to handle different coefficients and remainders. It also highlights the utility of synthetic division in evaluating functions, particularly for large numbers, and confirms the result using the Remainder Theorem. The host encourages viewers to subscribe and explore additional algebra and other math-related content available on the channel.

Takeaways
  • 📚 The video tutorial explains how to use synthetic division to divide polynomials.
  • 🔍 The first example involves dividing the polynomial x^3 + 3x^2 - 4x - 12 by x + 3, resulting in a quotient of x^2 - 4 and a remainder of zero.
  • 📝 The second example demonstrates dividing x^3 + x^2 - 2x - 8 by x - 2, yielding a quotient of x^2 + 3x + 4 with no remainder.
  • 🔢 The third example shows dividing x^3 + 5x^2 - 3x + 4 by x - 3, resulting in a quotient of x^2 + 8x + 21 and a remainder of 67.
  • ✂️ Synthetic division is used to evaluate functions by substituting a value into the polynomial and calculating the result.
  • 🔧 The remainder theorem is mentioned, which states that if a polynomial f(x) is divided by x - a, the remainder is f(a).
  • 📈 The video also shows how to use synthetic division to confirm the result of evaluating a function at a given value, like f(x) = 2x^3 - 3x^2 + 5x - 8 at x = 4.
  • 📉 The process of synthetic division involves bringing down the leading coefficient, multiplying by the divisor, adding, and repeating the process.
  • 📚 The video provides additional resources for those interested in more problems on synthetic division and long division.
  • 📺 The speaker encourages viewers to subscribe to the channel for more educational content.
  • 🎓 The channel offers a variety of educational videos on topics such as chemistry, physics, trigonometry, geometry, pre-calculus, and calculus.
Q & A

  • What is synthetic division?

    -Synthetic division is a method of dividing polynomials without the need for long division. It simplifies the process by using a specific sequence of operations involving multiplication and addition.

  • What is the first step in performing synthetic division?

    -The first step in synthetic division is to set the divisor polynomial equal to zero and solve for x to find the value that will be used in the division process.

  • How does the sign of the divisor affect the synthetic division process?

    -The sign of the divisor affects the synthetic division process because it determines the sign of the number that is used in the multiplication and addition steps. If the divisor is 'x - a', then 'a' is used as the number for synthetic division.

  • What does the remainder in synthetic division represent?

    -The remainder in synthetic division represents the value left over after the division process. If the remainder is zero, it indicates that the divisor is a factor of the dividend polynomial.

  • What is the result of dividing a cubic polynomial by a linear polynomial using synthetic division?

    -The result of dividing a cubic polynomial by a linear polynomial using synthetic division is a quadratic polynomial, provided that the remainder is zero.

  • How can synthetic division be used to evaluate functions?

    -Synthetic division can be used to evaluate functions by substituting the value of x into the polynomial and performing the division process to find the resulting value of the function.

  • What is the remainder theorem?

    -The remainder theorem states that if a polynomial f(x) is divided by x - a, the remainder of the division is equal to f(a). This theorem can be used to evaluate polynomials at specific points efficiently.

  • How can synthetic division be used to confirm the result of function evaluation?

    -Synthetic division can be used to confirm the result of function evaluation by dividing the polynomial by (x - a), where 'a' is the value at which the function is being evaluated, and comparing the result with the direct evaluation.

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

    -If the remainder is not zero after performing synthetic division, it indicates that the divisor is not a factor of the dividend polynomial, and the result of the division will be a quotient plus a remainder term.

  • Can synthetic division be used for polynomials of any degree?

    -Synthetic division can be used for polynomials of any degree, but it is typically used for dividing by linear factors (first-degree polynomials). For higher-degree divisors, long division or other methods may be more appropriate.

Outlines
00:00
📚 Introduction to Synthetic Division

This paragraph introduces the concept of synthetic division, a method used to divide polynomials. The example given is dividing the polynomial x^3 + 3x^2 - 4x - 12 by x + 3. The process involves setting x + 3 equal to zero to find the root, which is x = -3. The coefficients of the polynomial are then written down, and synthetic division is performed by bringing down the leading coefficient and successively multiplying and adding. The result is a quotient of x^2 - 4 and a remainder of zero, indicating that the division is exact.

05:00
🔍 Further Examples of Synthetic Division

The second paragraph continues the tutorial on synthetic division with additional examples. The first example divides x^3 + x^2 - 2x - 8 by x - 2, resulting in a quotient of x^2 + 3x + 4 with no remainder. The second example involves dividing x^3 + 5x^2 - 3x + 4 by x - 3, yielding a quotient of x^2 + 8x + 21 and a remainder of 67. This demonstrates that synthetic division can also be used to find remainders when the division is not exact.

10:01
📘 Evaluating Functions with Synthetic Division

The final paragraph of the script discusses the application of synthetic division in evaluating functions, particularly for large numbers. It uses the remainder theorem to evaluate the function f(x) = x^3 + 5x^2 - 3x + 4 at x = 3, confirming the remainder found in the previous example. The script then moves on to evaluate f(x) = 2x^3 - 3x^2 + 5x - 8 at x = 4, both by direct substitution and using synthetic division with x - 4, which confirms the same result of 92. The paragraph concludes by encouraging viewers to subscribe and explore additional video content on various mathematical topics.

Mindmap
Keywords
💡Synthetic Division
Synthetic division is a simplified method of polynomial division that is used to divide a polynomial by a binomial of the form x - c. In the video, synthetic division is the main technique discussed for dividing polynomials. It is demonstrated through various examples, starting with dividing a cubic polynomial by x + 3, and then by x - 2 and x - 3, showing the process of bringing down the leading coefficient, multiplying by c, and adding to find the next coefficient in the quotient.
💡Polynomial
A polynomial is an algebraic expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. The video script revolves around dividing polynomials, specifically cubic polynomials, using synthetic division. The examples given in the script involve polynomials such as x^3 + 3x^2 - 4x - 12, x^3 + x^2 - 2x - 8, and x^3 + 5x^2 - 3x + 4.
💡Coefficients
In the context of the video, coefficients are the numerical factors that multiply the variables in a polynomial. They are essential in synthetic division as they are the numbers that are manipulated to find the quotient and remainder. The script mentions coefficients like '1, 3, -4, -12' for the polynomial x^3 + 3x^2 - 4x - 12, which are used in the synthetic division process.
💡Remainder Theorem
The Remainder Theorem is a result in algebra that states if a polynomial f(x) is divided by x - c, the remainder of the division is f(c). In the script, the Remainder Theorem is illustrated by evaluating the function f(x) = 2x^3 - 3x^2 + 5x - 8 at x = 4 and confirming the result using synthetic division, showing that synthetic division can be used to evaluate functions, particularly at large numbers.
💡Quotient
The quotient is the result of division, specifically in polynomial division, it is the polynomial that remains after dividing the original polynomial by another polynomial. In the video, the quotient is obtained through synthetic division. For example, when dividing x^3 + 3x^2 - 4x - 12 by x + 3, the quotient is x^2 - 4 with a remainder of zero.
💡Remainder
In division, the remainder is what is left over after the division process if the divisor does not evenly divide the dividend. In the context of the video, the remainder is the result of the synthetic division process when the divisor does not divide the dividend exactly. For instance, when dividing x^3 + 5x^2 - 3x + 4 by x - 3, the remainder is 67.
💡Divisor
The divisor is the number or expression by which another number or expression is divided. In the script, the divisor is a binomial of the form x - c or x + c, such as x + 3, x - 2, and x - 3, which is used to divide the given polynomials through synthetic division.
💡Binomial
A binomial is a polynomial with two terms, typically expressed as ax + b. In the video, the divisors used in synthetic division are binomials, such as x + 3, x - 2, and x - 3. These binomials are crucial for setting up the synthetic division process.
💡Algebra
Algebra is a branch of mathematics that deals with symbols and the rules for manipulating these symbols. It is the broader mathematical context within which the video's content on synthetic division and polynomial division fits. The script mentions that the video is part of a larger series on algebra, covering various topics such as chemistry, physics, trigonometry, geometry, pre-calculus, and calculus.
💡Function Evaluation
Function evaluation is the process of finding the value of a function for a given input. In the video, function evaluation is demonstrated by calculating the value of the polynomial function f(x) = 2x^3 - 3x^2 + 5x - 8 at x = 4, both through direct substitution and using synthetic division to confirm the result, showing the practical application of synthetic division in evaluating functions.
Highlights

Introduction to synthetic division on polynomials.

Demonstration of synthetic division with the polynomial x^3 + 3x^2 - 4x - 12 divided by x + 3.

Explanation of how to find the root of the divisor to use in synthetic division.

Step-by-step guide on setting up synthetic division with coefficients.

Process of bringing down the leading coefficient in synthetic division.

Multiplication and addition steps in synthetic division explained.

Result of the first synthetic division example yielding a remainder of zero.

Interpretation of the result as x^2 - 4 when the remainder is zero.

Second example of synthetic division with the polynomial x^3 + x^2 - 2x - 8 divided by x - 2.

Change of sign for the divisor based on the root found.

Completion of the second synthetic division with a remainder of zero.

Result interpretation for the second example as x^2 + 3x + 4.

Third example with polynomial x^3 + 5x^2 - 3x + 4 divided by x - 3.

Illustration of synthetic division when a remainder is not zero.

Explanation of the result with a remainder as a fraction over the divisor.

Application of synthetic division to evaluate functions at specific values.

Demonstration of evaluating f(x) = 2x^3 - 3x^2 + 5x - 8 at x = 4.

Confirmation of function evaluation using synthetic division.

Introduction to the remainder theorem and its relation to synthetic division.

Invitation to subscribe and explore additional video content on various mathematical topics.

Transcripts

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Synthetic DivisionPolynomialsAlgebraEducationalMath TutorialRemainder TheoremFunction EvaluationMathematicsLearning ResourceProblem Solving