Polynomial Long Division and Remainder Theorem

The video introduces the Polynomial Remainder Theorem, explaining its initial magical feel and promising a proof in future videos. The theorem states that the remainder of a polynomial f(x) divided by x-a is f(a). An example with a second-degree polynomial is used to illustrate the theorem, followed by a detailed polynomial long division. The remainder is verified to match f(a), demonstrating the theorem's validity. The video concludes by highlighting the theorem's utility in quickly finding remainders without full division.

What does the Polynomial Remainder Theorem state about the remainder when a polynomial f(x) is divided by x-a?

• The remainder is the derivative of f(x).

• The remainder is the integral of f(x).

• The remainder is f(a).

• The remainder is always zero.

In the given example, what is the value of 'a' when dividing the polynomial by x-a?

• 2

• 0

• 1

• 3

What is the first step in performing polynomial long division?

• Divide the constant terms.

• Subtract the polynomials.

• Add the polynomials.

• Multiply the divisor by the leading term of the dividend.

What is the purpose of subtracting in polynomial long division?

• To simplify the polynomial.

• To find the remainder.

• To find the quotient.

• To eliminate the leading term.

What is the remainder when 3x^2 - 4x + 7 is divided by x-1?

• 5

• 6

• 7

• 8

What is the degree of the remainder in polynomial long division?

• Equal to the degree of the divisor.

• Higher than the degree of the divisor.

• Lower than the degree of the divisor.

• Equal to the degree of the dividend.

What is the result of evaluating f(1) for the polynomial 3x^2 - 4x + 7?

• 8

• 7

• 5

• 6

How can the Polynomial Remainder Theorem simplify finding the remainder of a polynomial division?

• By eliminating the need for division.

• By using the derivative of the polynomial.

• By integrating the polynomial.

• By evaluating the polynomial at a specific point.

Why is the Polynomial Remainder Theorem considered useful?

• It is used in integration.

• It helps in finding the roots of the polynomial.

• It simplifies finding the remainder without full division.

• It provides the exact quotient.

What is the main focus of the Polynomial Remainder Theorem?

• Finding the derivative.

• Finding the roots.

• Finding the remainder.

• Finding the quotient.