Factoring and Solving Polynomial Equations

This video tutorial explains how to solve cubic equations using the factor by grouping technique. It provides two examples: the first involves solving x^3 + 2x^2 - 9x - 18 = 0, and the second involves solving x^3 - x^2 - 3x + 15 = 0. The tutorial demonstrates factoring by grouping, identifying common binomial factors, and applying the zero product property to find solutions. The first example results in three real rational solutions, while the second example yields one real rational solution and two real irrational solutions.

What technique is recommended for solving cubic equations with four terms?

• Quadratic formula

• Synthetic division

• Completing the square

• Factor by grouping

What is the greatest common factor of the first two terms in the equation x^3 + 2x^2 - 9x - 18?

• x

• x^2

• 2x

• 9

Why do we factor out negative nine instead of positive nine in the first example?

• To eliminate negative terms

• To make the equation positive

• To match the common binomial factor

• To simplify the equation

What is the result of factoring x^2 - 9 in the first example?

• x + 2 and x - 2

• x + 3 and x - 3

• x^2 + 9

• x^2 - 3

What are the solutions for the equation x^3 + 2x^2 - 9x - 18 = 0?

• x = 2, x = -3, x = 3

• x = -2, x = -3, x = 3

• x = 2, x = 3, x = -3

• x = -2, x = 3, x = -3

What is the greatest common factor of the first two terms in the second example equation?

• 5x

• x

• x^2

• 3

Why do we factor out negative three instead of positive three in the second example?

• To make the equation positive

• To eliminate negative terms

• To match the common binomial factor

• To simplify the equation

What method is used to solve x^2 - 3 = 0 in the second example?

• Completing the square

• Quadratic formula

• Square root property

• Synthetic division

What are the solutions for the equation x^3 - x^2 - 3x + 15 = 0?

• x = 5, x = ±√3

• x = 5, x = ±3

• x = -5, x = ±3

• x = -5, x = ±√3

How many real irrational solutions are there in the second example?

• One

• Two

• Three

• None