Polynomial remainder theorem example | Polynomial and rational functions | Algebra II | Khan Academy

TL;DR

The Polynomial Remainder Theorem allows for a simpler way to find the remainder when dividing a polynomial by a given expression.

Transcript

• So we have a polynomial here. What I'm curious about is what is the remainder if I were to divide this polynomial by, let's just say, x minus, I want the remainder when I divide this polynomial by x minus two? You could do this. You could figure this out with algebraic long division, but I'll give you a hint. It is much simpler and much less comp... Read More

Key Insights

• ❓ The Polynomial Remainder Theorem simplifies finding polynomial remainders by evaluating the polynomial at a given value.
• 👾 Algebraic long division is a more computationally intensive and space-consuming method compared to using the Polynomial Remainder Theorem.
• 🈸 The degree of the polynomial doesn't affect the application of the Polynomial Remainder Theorem.
• 💨 The Polynomial Remainder Theorem provides a faster and more efficient way to find remainders, especially when dealing with larger polynomials.

Questions & Answers

Q: What is the Polynomial Remainder Theorem?

The Polynomial Remainder Theorem states that when dividing a polynomial by (x-a), the remainder is equal to the value of the polynomial when evaluated at a. It provides a simpler method to find remainders compared to algebraic long division.

Q: How can the remainder be found using the Polynomial Remainder Theorem?

To find the remainder when dividing a polynomial by (x-a), evaluate the polynomial at the value of "a". The result will be the remainder.

Q: What is the benefit of using the Polynomial Remainder Theorem?

The Polynomial Remainder Theorem offers a simpler and less computation-intensive method to find remainders, eliminating the need for lengthy algebraic long division calculations. It saves time and space on paper.

Q: Can the Polynomial Remainder Theorem be used for polynomials of any degree?

Yes, the Polynomial Remainder Theorem can be applied to polynomials of any degree. It provides a general method to find remainders when dividing polynomials by expressions of the form (x-a).

Summary & Key Takeaways

• The Polynomial Remainder Theorem states that when dividing a polynomial by (x-a), the remainder is equal to the value of the polynomial when evaluated at a.

• In this case, the polynomial is (-3x^2 + 4x -7), and when dividing it by (x-2), the remainder can be found by evaluating the polynomial at x=2.

• By using the Polynomial Remainder Theorem, the remainder (-27) can be determined without the need for algebraic long division.