Understanding the Factor Theorem and Solving Polynomial Equations

The video tutorial focuses on using the factor theorem to solve polynomial equations. It begins by explaining the factor theorem, which states that if f(c) = 0, then x-c is a factor of f(x). The video then demonstrates solving polynomial equations using synthetic division, providing examples and step-by-step instructions. It also covers how to factor polynomials completely and find all zeros of the function. The tutorial concludes with solving a polynomial equation given one of its roots, using the factor theorem and synthetic division.

What does the Factor Theorem state about the relationship between f(c) and x - c?

• If f(c) = 0, then x - c is a factor of f(x).

• If f(c) = 1, then x - c is a factor of f(x).

• If f(c) = 0, then x - c is not a factor of f(x).

• If f(c) = 0, then x + c is a factor of f(x).

In synthetic division, what is the first step after writing down the coefficients?

• Add the divisor to the first coefficient.

• Multiply the divisor by the first coefficient.

• Subtract the divisor from the first coefficient.

• Bring down the first coefficient.

What is the result of dividing x^3 - 4x^2 + x + 6 by x - 3 using synthetic division?

• x^2 + 2x - 1

• x^2 - 2x + 1

• x^2 - x - 2

• x^2 + x + 2

How do you determine the zeros of a polynomial after factorization?

• Set each factor equal to one.

• Set each factor equal to zero.

• Multiply each factor by zero.

• Add each factor to zero.

Given a root of a polynomial, what is the first step in using synthetic division?

• Add the root to the polynomial.

• Subtract the root from the polynomial.

• Multiply the root by the polynomial.

• Use the root as the divisor in synthetic division.

What is the factor corresponding to the root x = -3/2?

• x + 3/2

• 2x + 3

• x - 3/2

• 2x - 3

What is the GCF taken out from the expression 12x^2 - 2x - 2?

• 2

• 3

• 12

• 6

Which two numbers multiply to -6 and add to -1 in the factorization process?

• -2 and 3

• 1 and -6

• 2 and -3

• -3 and 2

What is the final step in solving a polynomial equation after factorization?

• Divide all factors by zero.

• Add all factors together.

• Set each factor equal to zero and solve for x.

• Multiply all factors together.

What are the roots of the polynomial 12x^3 + 16x^2 - 5x - 3?

• -3/2, 1/2, -1/3

• -3/2, -1/2, 1/3

• 3/2, 1/2, -1/3

• 3/2, -1/2, 1/3