Polynomial remainder theorem to test factor | Algebra II | Khan Academy | Summary and Q&A

TL;DR

Evaluating the remainder of a polynomial division by a first degree expression helps determine if it is a factor.

Key Insights

• 😑 Algebraic long division can be used to determine if an expression is a factor of a polynomial.
• ➗ The polynomial remainder theorem helps calculate the remainder quickly for first degree polynomial divisions.
• 0️⃣ Zero remainder indicates that the expression is a factor, while non-zero remainder indicates it is not.

Transcript

Read and summarize the transcript of this video on Glasp Reader (beta).

Questions & Answers

Q: How can we determine if an expression is a factor of a polynomial?

To determine if an expression is a factor, we can divide the polynomial by that expression and check if there is a remainder. A remainder of zero indicates that the expression is a factor.

Q: What is the polynomial remainder theorem?

The polynomial remainder theorem states that if we divide a polynomial by a first degree expression (like "x minus a"), the remainder will be equal to the polynomial evaluated at the value of "a." This theorem helps us calculate the remainder quickly.

Q: What happens if there is a non-zero remainder after polynomial division?

If there is a non-zero remainder after dividing the polynomial by an expression, it means that the expression is not a factor of the polynomial. The presence of a remainder indicates that the polynomial cannot be fully divided by the expression.

Q: How can we evaluate the remainder of a polynomial division?

To evaluate the remainder, substitute the value of the variable in the expression being divided by into the polynomial. Calculate the value of the polynomial at that point to obtain the remainder of the division.

Summary & Key Takeaways

• The video explains how to determine if the expression "x minus three" is a factor of a given fourth-degree polynomial.

• Algebraic long division can be used to divide the polynomial by "x minus three" and check for a remainder.

• If the remainder is zero, "x minus three" is a factor; if there is a non-zero remainder, it is not a factor.