Tamara J. answered • 02/20/13

Math Tutoring - Algebra and Calculus (all levels)

f(x) = x^{3} + 3x^{2} - x - 3

(1) To factor a polynomial equation such as this one, consider the first two terms and last terms separately:

f(x) = (x^{3} + 3x^{2}) + (-x - 3)

Factor out the greatest common factor (gcf) from the first two terms and the gcf from the last two terms:

f(x) = (x^{2})(x + 3) + (-1)(x + 3)

Notice that these two terms together have a gcf that you can factor out. That being, x + 3:

f(x) = (x + 3)((x^{2}) + (-1))

= (x + 3)(x^{2} - 1)

The last term is a perfect square and can, therefore, be factored out further:

x^{2} - 1 = (x + 1)(x - 1)

That is,

f(x) = (x + 3)(x^{2} - 1)

**f(x) = (x + 3)(x + 1)(x - 1)**

(2) To find the zeros of this equation, set the equation equal to 0:

(x + 3)(x + 1)(x - 1) = 0

By the zero product rule, we can set each term equal to 0 and solve for x:

x + 3 = 0 ==> x + 3 - 3 = 0 - 3 ==> x = -3

x + 1 = 0 ==> x + 1 - 1 = 0 - 1 ==> x = -1

x - 1 = 0 ==> x - 1 + 1 = 0 + 1 ==> x = 1

Thus, the zeros for this equation are: **x = -3** , **x = -1** , ** x = 1**

The y-intercept is the point at which the graph of this equation crosses the y-axis. That is, the y-intercept is the point at which x=0. We do this by solving for y=f(x) when x=0; or, similarly, solve for y=f(0):

y = f(x) = x^{3} + 3x^{2} - x - 3

y = f(0) = (0)^{3} + 3(0)^{2} - 0 - 3

= 0 + 0 - 0 - 3

= -3

**y = f(0) = -3**

Thus, the y-intercept is at y = -3; or, the point (0, -3).