Understanding Polynomial Factors and the Remainder Theorem

The video tutorial explains how to determine the value of c for which x-3 is a factor of the polynomial p(x) = -x^3 + cx^2 - 4x + 3. Using the remainder theorem, it is shown that p(3) must equal zero for x-3 to be a factor. The tutorial walks through evaluating p(3) and solving the resulting equation to find that c equals 4.

What is the main goal of the problem introduced in the video?

• To determine the value of c that makes x-3 a factor of p(x).

• To calculate the derivative of p(x).

• To find the roots of the polynomial p(x).

• To find the value of x for which p(x) is zero.

According to the Remainder Theorem, when is x-3 a factor of p(x)?

• When p(3) is undefined.

• When p(3) equals zero.

• When p(3) is less than zero.

• When p(3) is greater than zero.

What does the Remainder Theorem imply about the expression p(x) when x-3 is a factor?

• p(x) is a constant value.

• p(x) is an exponential function.

• p(x) can be expressed as x-3 times another polynomial.

• p(x) is a quadratic equation.

What is the first step in evaluating p(3) in the video?

• Graph the polynomial.

• Integrate the polynomial.

• Differentiate the polynomial.

• Substitute 3 into the polynomial and simplify.

What is the expression obtained after substituting 3 into p(x)?

• 9c - 36

• 9c + 36

• 27c - 36

• 27c + 36

What equation is set up to find the value of c?

• c + 36 = 0

• 9c + 36 = 0

• 9c - 36 = 0

• c - 36 = 0

What is the value of c that makes x-3 a factor of p(x)?

• c = 3

• c = 4

• c = 5

• c = 6

What operation is performed to isolate c in the equation 9c = 36?

• Subtract 9 from both sides.

• Multiply both sides by 9.

• Divide both sides by 9.

• Add 9 to both sides.

What is the final form of the polynomial p(x) when c is found?

• p(x) = -x^3 + 6x^2 - 4x + 3

• p(x) = -x^3 + 5x^2 - 4x + 3

• p(x) = -x^3 + 4x^2 - 4x + 3

• p(x) = -x^3 + 3x^2 - 4x + 3

What conclusion is reached about the polynomial when c equals 4?

• x-3 is not a factor of p(x).

• p(x) is a constant.

• x-3 is a factor of p(x).

• p(x) has no factors.