Polynomial Equations Quiz

This tutorial covers solving polynomial equations, focusing on techniques like factoring by grouping, substitution, and using the rational zero theorem with synthetic division. It includes examples of solving cubic and quartic polynomials, highlighting the use of imaginary numbers and the difference of perfect squares. The video aims to equip learners with methods to find real and imaginary solutions for polynomial equations.

What is the first step in solving the equation x^3 + 3x^2 - 4x - 12 = 0 using factor by grouping?

• Divide the equation by x

• Use the quadratic formula

• Find the greatest common factor of the first two terms

• Add 12 to both sides

When reducing a quartic equation to a quadratic form, what substitution is typically used?

• a = x^3

• a = x^4

• a = x^2

• a = x

Which two numbers multiply to -36 and add to -5 in the equation x^4 - 5x^2 - 36 = 0?

• -6 and 6

• -9 and 4

• 9 and -4

• 6 and -6

What is the first step in applying the Rational Zero Theorem to find zeros of a polynomial?

• Apply the quadratic formula

• Use synthetic division

• List factors of the leading coefficient

• List factors of the constant term

In the equation x^3 - 3x^2 - 6x + 8 = 0, what is the result of evaluating the function at x = 1?

• 2

• -1

• 1

• 0

What is the next step after finding one zero using the Rational Zero Theorem?

• Factor by grouping

• Add the zero to the equation

• Use the quadratic formula

• Perform synthetic division

In the equation x^4 - 7x^3 + 7x^2 + 35x - 60 = 0, which number is a zero of the polynomial?

• 3

• 2

• 4

• 1

What is the result of synthetic division when x = 3 is a zero of the polynomial x^4 - 7x^3 + 7x^2 + 35x - 60?

• x^3 - 4x^2 - 5x + 20

• x^3 + 4x^2 + 5x - 20

• x^3 - 4x^2 + 5x - 20

• x^3 + 4x^2 - 5x + 20

What technique is used to factor x^2 - 5 in the equation x^4 - 7x^3 + 7x^2 + 35x - 60?

• Rational Zero Theorem

• Quadratic formula

• Difference of perfect squares

• Factor by grouping

Which method is NOT mentioned as a way to solve polynomial equations in the conclusion?

• Substitution

• Rational Zero Theorem

• Completing the square

• Factor by grouping