Understanding the Remainder Theorem

The video tutorial explains the remainder theorem, which states that if a polynomial f(x) is divided by x-k, the remainder is f(k). The instructor demonstrates this using synthetic division and evaluates the polynomial to show that the remainder is zero when the divisor is a factor. The video also highlights the importance of the theorem in determining factors of polynomials without division.

What does the Remainder Theorem state?

• If a polynomial is divided by a linear factor, the remainder is the value of the polynomial evaluated at the root of the factor.

• If a polynomial is divided by a linear factor, the remainder is always zero.

• If a polynomial is divided by a linear factor, the remainder is the value of the polynomial evaluated at zero.

• If a polynomial is divided by a quadratic factor, the remainder is the value of the polynomial evaluated at the root of the factor.

In the example provided, what are the coefficients used for synthetic division?

• 5, -7, 24

• 5, 7, -24

• 5, -7, -24

• 5, 7, 24

What is the resulting factor when the polynomial is divided by x + 3?

• 5x + 8

• 5x - 8

• 5x - 3

• 5x + 3

What is the final step in synthetic division?

• Drop down the first coefficient.

• Subtract the last coefficient from the previous result.

• Add the last coefficient to the previous result.

• Multiply the last coefficient by the divisor.

What is the value of f(-3) for the polynomial 5x^2 + 7x - 24?

• 0

• 10

• -10

• 24

What is the result of 5 times (-3)^2 in the verification example?

• 15

• 30

• 45

• 60

Why is the Remainder Theorem important?

• It helps in finding the degree of a polynomial.

• It helps in simplifying polynomials.

• It helps in determining if a factor evenly divides a polynomial.

• It helps in finding the roots of a polynomial.

What is the purpose of evaluating a polynomial at a specific value according to the Remainder Theorem?

• To find the constant term of the polynomial.

• To find the remainder when the polynomial is divided by a linear factor.

• To determine the leading coefficient.

• To find the degree of the polynomial.

What is the value of f(2) for the polynomial 5x^2 + 7x - 24?

• 24

• 34

• 10

• 0

What is the remainder when the polynomial 5x^2 + 7x - 24 is divided by x - 2?

• 0

• 10

• 34

• 24