Factoring and Solving Polynomial Equations

The video tutorial explains how to solve the equation x^3 - 49x = 0 by factoring. It begins by factoring out the greatest common factor, x, and then recognizing the difference of squares in x^2 - 49. The equation is factored completely into x(x + 7)(x - 7) = 0. Using the zero product property, the solutions x = 0, x = -7, and x = 7 are found. The tutorial concludes by graphically verifying these solutions by identifying the x-intercepts of the graph of y = x^3 - 49x.

What is the first step in factoring the equation x^3 - 49x = 0?

• Identify the difference of squares

• Factor out the greatest common factor

• Apply the quadratic formula

• Complete the square

What is the greatest common factor in the equation x^3 - 49x?

• x

• x^2

• 7

• 49

How can x^2 - 49 be factored further?

• As a perfect square trinomial

• As a linear binomial

• As a difference of squares

• As a sum of cubes

What are the factors of x^2 - 49?

• (x + 7)(x - 7)

• (x - 7)^2

• (x + 49)(x - 49)

• (x + 7)^2

What property is used to find the solutions of the equation after factoring?

• Commutative Property

• Associative Property

• Zero Product Property

• Distributive Property

What are the solutions to the equation x^3 - 49x = 0?

• x = 0, x = 49

• x = 0, x = 7, x = -7

• x = 0, x = 49, x = -49

• x = 7, x = -7

How can the solutions of the equation be verified graphically?

• By finding the maximum point

• By finding the y-intercepts

• By finding the x-intercepts

• By finding the slope of the graph

What does the point (0, 0) on the graph of y = x^3 - 49x represent?

• A maximum point

• A minimum point

• An x-intercept

• A y-intercept

Which ordered pair represents a solution to the equation on the graph?

• (49, 0)

• (0, 7)

• (0, 49)

• (7, 0)

What does the point (-7, 0) on the graph indicate?

• A point of inflection

• A local minimum

• A local maximum

• A solution to the equation