Polinomial (Bagian 4) - Teorema Sisa dan Teorema Faktor

m4th-lab

2 Feb 202116:07

Summary

TLDRThe video script is a comprehensive tutorial on polynomial division and factor theorems. It introduces the Remainder Theorem and Factor Theorem, explaining how to find the remainder of a polynomial division without performing the actual division process. The script provides step-by-step examples, including substituting values to simplify calculations. It also covers how to determine factors of a polynomial when the remainder is zero upon division. The tutorial aims to make polynomial division more efficient and understandable for viewers.

Takeaways

📚 The video is part of a series discussing polynomials, specifically focusing on the fourth part which covers the Remainder Theorem and Factor Theorem.
🔍 The Remainder Theorem states that the remainder of the division of a polynomial f(x) by x - k is f(k), and the video explains how to find this remainder without full division.
📝 An example is provided to illustrate the Remainder Theorem, showing the process of substituting the value that makes the divisor zero into the polynomial to find the remainder.
📉 The video also explains the second form of the Remainder Theorem, which involves dividing by x + b and finding the remainder by substituting -b into the polynomial.
🔢 A detailed example is given for dividing a cubic polynomial by a linear binomial, demonstrating the step-by-step process of finding the remainder.
📈 The third form of the Remainder Theorem is introduced for dividing by a quadratic polynomial, resulting in a remainder that is a first-degree polynomial.
🔑 The Factor Theorem is discussed, which states that a polynomial f(x) is a factor of another polynomial if the remainder is zero when f(x) is divided by it.
🌰 An example polynomial is factored using the Factor Theorem, showing that x - 1 is a factor and then finding the other factor by dividing the polynomial by x - 1.
📚 The video concludes with a brief mention of the next topic, which will be polynomial equations, indicating a continuation of the series.
👋 The video ends with a sign-off greeting, wishing the viewers well in Arabic, which is a common practice in educational content to maintain cultural relevance.

Q & A

What is the main topic discussed in the fourth part of the polynomial series video?
-The main topic discussed in the fourth part of the polynomial series video is the Remainder Theorem and Factor Theorem.
What is the Remainder Theorem in the context of the video?
-The Remainder Theorem, as discussed in the video, is a method to find the remainder of a polynomial division without performing the actual division process.
How can one find the remainder of a polynomial division using the Remainder Theorem?
-To find the remainder of a polynomial division using the Remainder Theorem, one can substitute the value that makes the divisor zero into the polynomial and evaluate it.
What is an example of using the Remainder Theorem as shown in the video?
-An example given in the video is to find the remainder of the polynomial \( x^4 - 2x^3 + 4x^2 - 5 \) divided by \( x - 1 \). By substituting \( x = 1 \) into the polynomial, the remainder is found to be -2.
What is the second Remainder Theorem mentioned in the video?
-The second Remainder Theorem mentioned in the video states that the remainder of a polynomial \( f(x) \) divided by \( x + k \) is \( f(-k) \).
Can you provide an example of the second Remainder Theorem from the video?
-An example from the video is to find the remainder of the polynomial \( x^3 - 2x^2 + 3 \) divided by \( 2x + 3 \). By setting \( 2x + 3 = 0 \) and solving for \( x \), we get \( x = -\frac{3}{2} \). Substituting this value into the polynomial gives the remainder.
What is the third Remainder Theorem discussed in the video?
-The third Remainder Theorem discussed in the video is for dividing a polynomial by a quadratic polynomial, where the remainder is a linear polynomial \( S = P(x) + Q \) with \( Fa = p + q \) and \( Fb = pb \) plus a constant \( K \).
How does the Factor Theorem relate to the Remainder Theorem?
-The Factor Theorem is closely related to the Remainder Theorem. A polynomial \( f(x) \) is a factor of another polynomial if the remainder is zero when the latter is divided by the former.
What is an example of using the Factor Theorem from the video?
-An example from the video is the polynomial \( x^3 + x^2 + x - 3 \). By testing \( x = 1 \) and finding that the remainder is zero, it is concluded that \( x - 1 \) is a factor of the polynomial.
What is the significance of the Remainder Theorem and Factor Theorem in polynomial division?
-The Remainder Theorem and Factor Theorem are significant as they provide a quick way to determine the remainder of a polynomial division and to identify factors of a polynomial without performing long division.
What is the next topic to be covered after the Remainder and Factor Theorems in the polynomial series video?
-The next topic to be covered after the Remainder and Factor Theorems in the polynomial series video is polynomial equations.

Outlines

00:00

📚 Introduction to Polynomial Division Theorems

This paragraph introduces the topic of polynomial division, specifically focusing on the Remainder Theorem and Factor Theorem. The speaker, Dini, encourages viewers to watch previous videos for foundational knowledge and then explains the concept of the remainder when a polynomial is divided by another polynomial. The Remainder Theorem is introduced as a method to find the remainder of a polynomial division without performing the actual division process. An example is given to illustrate how to apply the theorem by substituting the divisor into the polynomial to find the remainder.

05:03

🔍 Detailed Explanation of Remainder Theorems

The second paragraph delves deeper into the Remainder Theorems, explaining two variations. The first theorem is reiterated with an example of dividing a polynomial by \( x - 1 \) and finding the remainder by substituting \( x = 1 \) into the polynomial. The second theorem is introduced with an example of dividing by \( 2x + 3 \), where the process involves finding the zero of the divisor and substituting it into the polynomial to calculate the remainder. The explanation includes step-by-step calculations and simplifications to arrive at the remainder.

10:06

📘 Polynomial Division by Quadratics and Factor Theorem

This paragraph discusses the third Remainder Theorem, which applies when a polynomial is divided by a quadratic expression. The theorem states that the remainder will be a polynomial of degree one less than the divisor. An example problem is presented where given remainders from divisions by \( x + 2 \) and \( x - 4 \), the task is to find the remainder when the polynomial is divided by \( x^2 - 2x - 8 \). The solution involves setting up equations based on the given remainders and solving for the unknown coefficients using the Remainder Theorem. The paragraph concludes with finding the specific values of the coefficients and the remainder.

15:08

🔑 Factor Theorem and Identifying Polynomial Factors

The final paragraph introduces the Factor Theorem, which is used to determine if a polynomial is a factor of another polynomial by checking if the remainder is zero when the polynomial is divided by the potential factor. The theorem is illustrated with an example where the polynomial \( x^3 + x^2 - x - 3 \) is tested for divisibility by \( x - 1 \). The process involves substituting \( x = 1 \) into the polynomial and verifying that the result is zero, confirming \( x - 1 \) as a factor. The paragraph concludes with a brief mention of the next topic, polynomial equations, to be covered in subsequent videos.

Mindmap

Keywords

💡Polynomial

A polynomial is a mathematical expression involving a sum of powers in a single variable. In the video's theme, polynomials are the central objects of study, with discussions on division, remainders, and factors. For example, the script mentions 'polinomial x pangkat 4 min dua x pangkat 3 ditambah 4 x kuadrat min 5', which translates to 'the polynomial x to the power of 4 minus 2 times x to the power of 3 plus 4 times x squared minus 5'.

💡Remainder Theorem

The Remainder Theorem states that if a polynomial is divided by a linear divisor of the form 'x - c', the remainder is the value of the polynomial evaluated at x = c. The video explains this theorem by demonstrating how to find the remainder without performing long division, as seen in the script's example of dividing 'x^4 - 2x^3 + 4x^2 - 5' by 'x - 1'.

💡Factor Theorem

The Factor Theorem is a specific case of the Remainder Theorem, stating that a polynomial has a factor 'x - c' if and only if the remainder is zero when the polynomial is divided by 'x - c'. The video connects this theorem to the concept of factors in polynomials, illustrating it with the example of 'x^3 + x^2 + x - 3' being divisible by 'x - 1' because the remainder is zero when x = 1.

💡Division

In the context of the video, division refers to the process of dividing one polynomial by another, aiming to find a quotient and possibly a remainder. The script discusses efficient methods to find remainders without full division, such as using the Remainder Theorem, which simplifies the process as shown in the example of dividing 'x^3 - 2x^2 + 3' by '2x + 3'.

💡Quotient

The quotient is the result of division when the dividend is exactly divisible by the divisor. In polynomial division, the quotient is another polynomial. The video script mentions finding the quotient in the context of using the Remainder Theorem to simplify the division process, as in the example where 'x^4 - 2x^3 + 4x^2 - 5' divided by 'x - 1' yields a quotient and a remainder.

💡Polynomial Degree

The degree of a polynomial is the highest power of the variable in the polynomial. The video emphasizes the degree in relation to the Remainder Theorem and Factor Theorem, explaining that the remainder's degree is always less than the divisor's degree, as illustrated in the script when discussing dividing a quartic polynomial by a linear one.

💡Substitution

Substitution is the process of replacing the variable in an expression with a specific value. The video script uses substitution to apply the Remainder Theorem, where the value of the divisor's variable (x) is substituted with the value that makes the divisor zero, as seen in the example where 'x' is substituted with '1' to find the remainder of the division.

💡Zeroes of a Polynomial

The zeroes of a polynomial are the values of the variable that make the polynomial equal to zero. The video script refers to finding zeroes when applying the Remainder Theorem, as the theorem involves setting the polynomial's divisor to zero to find the remainder, such as setting 'x + 2' to zero to find the remainder when dividing by 'x + 2'.

💡Factorization

Factorization in the context of polynomials is the process of breaking down a polynomial into a product of other polynomials. The video script discusses factorization as part of finding all factors of a polynomial, using the Factor Theorem to identify 'x - 1' as a factor and then dividing the original polynomial by this factor to find the other factors.

💡Horner's Method

Horner's Method, also known as the Horner's scheme, is an efficient algorithm for evaluating polynomials. The video script briefly mentions using Horner's Method for polynomial division, which simplifies the process by transforming the polynomial into a nested form and then evaluating it step by step, as seen in the script's reference to evaluating 'x^3 + x^2 + x - 3' using this method.

Highlights

Introduction to the fourth part of the polynomial series video, focusing on the Remainder Theorem and Factor Theorem.

Explanation of the Remainder Theorem, stating that the remainder of a polynomial divided by (x - a) is equal to the polynomial evaluated at x = a.

Demonstration of how to find the remainder of a polynomial division without performing the actual division process.

Use of the Remainder Theorem to simplify the process of finding remainders, making it faster and more efficient.

Example problem: Finding the remainder of the polynomial \( x^4 - 2x^3 + 4x^2 - 5 \) when divided by \( x - 1 \).

Step-by-step substitution of x = 1 into the polynomial to find the remainder.

Introduction of the second Remainder Theorem, which deals with dividing by (x + a).

Method to find the remainder when dividing by (x + a) by substituting the value that makes the divisor zero.

Example problem: Determining the remainder of the polynomial \( x^3 - 2x^2 + 3 \) when divided by \( 2x + 3 \).

Explanation of the third Remainder Theorem, which involves dividing by a quadratic polynomial of the form (x - a)(x - b).

How to find the remainder when dividing by a quadratic polynomial by using the first two Remainder Theorems.

Example problem: Finding the remainder of a polynomial when divided by \( x^2 - 2x - 8 \) using previously found remainders.

Use of algebraic manipulation to solve for the unknown coefficients in the remainder polynomial.

Introduction to the Factor Theorem, which states that a polynomial has a factor if the remainder is zero when divided by that factor.

Illustration of the Factor Theorem with an example polynomial and its factors.

Method to determine additional factors of a polynomial by dividing the polynomial by a known factor.

The use of Horner's method for efficient polynomial division to find factors.

Conclusion of the video with a summary of the three Remainder Theorems and the Factor Theorem.

Announcement of the next video in the series, which will cover polynomial equations.