Polinomial (Bagian 2) - Menentukan Nilai Polinomial dengan Substitusi dan Skema Horner

Summary

This video from the math-lab channel, hosted by Deni and Handayani, explores the methods of determining polynomial values. The script introduces two techniques: substitution and Horner's scheme. Through examples, including a parabolic trajectory of a basketball, the video demonstrates how to substitute a variable's value directly into the polynomial equation and how to use Horner's method for efficient computation. The educational content is designed to help viewers understand and compare these two approaches in solving polynomial equations.

Takeaways

• 📚 This video is about polynomial (or many-term) concepts and focuses on determining polynomial values using two methods: substitution and Horner's scheme.
• 🏀 The example given involves a basketball's parabolic trajectory described by the equation h = -4t^2 + 5t + 2, where h is the height in meters and t is the time in seconds.
• ⏱️ To find the height of the ball at t = 1, the substitution method is used first, replacing t with 1 in the equation, resulting in a height of 3 meters.
• 🔄 The substitution method involves directly substituting the value of the variable into the polynomial equation.
• 🔢 Horner's scheme is introduced as an alternative method for evaluating polynomials, using a more systematic approach that involves coefficients and constants.
• 📊 In Horner's scheme, the coefficients and constants of the polynomial are written in order, and a systematic process of multiplication and addition is followed to find the polynomial's value.
• ✍️ Using Horner's scheme for the same polynomial h = -4t^2 + 5t + 2 at t = 1 also results in a height of 3 meters, confirming the consistency of both methods.
• 🧮 A more complex example is provided with f(x) = 3x^4 - 2x^3 + x - 3, showing the step-by-step application of Horner's scheme to find f(2) = 31.
• 📝 Another example, f(x) = 3x^6 + 7x^3 + 8x + 10, demonstrates finding f(-2) = 130 using Horner's scheme, illustrating the method's adaptability to different polynomials.
• 🔍 The video concludes with a final example and a practice problem, encouraging viewers to apply both substitution and Horner's scheme to ensure understanding and accuracy.

Q & A

What is the main topic of the video?

The main topic of the video is how to determine the value of a polynomial using two methods: substitution and Horner's scheme.

What example is used to illustrate polynomial evaluation?

The example used is the height of a basketball's trajectory, given by the polynomial h(t) = -4t^2 + 5t + 2, where h is the height in meters and t is the time in seconds.

How is the height of the basketball at t = 1 second calculated using substitution?

The height is calculated by substituting t = 1 into the polynomial h(t) = -4t^2 + 5t + 2, resulting in h(1) = -4(1)^2 + 5(1) + 2 = 3 meters.

What is the first step in evaluating a polynomial using Horner's scheme?

The first step is to list the coefficients of the polynomial in descending order of their corresponding powers.

How are coefficients used in Horner's scheme?

The coefficients are used in a series of multiplications and additions to simplify the polynomial evaluation process.

In the example with the polynomial h(t) = -4t^2 + 5t + 2, what are the coefficients used in Horner's scheme?

The coefficients are -4, 5, and 2.

What is the result of evaluating the polynomial h(t) = -4t^2 + 5t + 2 at t = 1 using Horner's scheme?

The result is h(1) = 3 meters, the same as with the substitution method.

Why might one prefer Horner's scheme over substitution?

Horner's scheme can be more efficient and less prone to errors for higher-degree polynomials, especially when evaluating them at multiple points.

What additional example is provided to demonstrate Horner's scheme?

An additional example is evaluating the polynomial f(x) = 3x^4 - 2x^3 + x - 3 at x = 2.

What is the final value of the polynomial f(x) = 3x^4 - 2x^3 + x - 3 at x = 2 using Horner's scheme?

The final value is f(2) = 31.