Can you help me?

Question

I really dont understand how to do any of these things...28 in Algebra, ugh HELP (teacher is no good). ive been given f(x)=x^3+3x^2-x-3. i need to factor, find zeros and find y-intercept.

Answer 1

     f(x) = x³ + 3x² - x - 3

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

          f(x) = (x³ + 3x²) + (-x - 3)

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

          f(x) = (x²)(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)((x²) + (-1))

                = (x + 3)(x² - 1)

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

      x² - 1 = (x + 1)(x - 1)

That is, 

           f(x) = (x + 3)(x² - 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) = x³ + 3x² - 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).

 

Answer 2

Factoring polynomials is a matter of recognizing patterns, that is, common factors, and that takes practice, practice, practice.

Here are two ways to factor this polynomial:

f(x) = x³ + 3x² – x – 3
      = (x + 3) x² – 1 * (x + 3)                     recognize (x + 3) as a common factor
      = (x + 3) (x² – 1)                                factor (x + 3) out of both parts
      = (x + 3) (x + 1) (x – 1)                       a² + b² = (a + b) (a - b)

or
f(x) = x³ + 3x² – x – 3
      = (x³ – x) + (3x² – 3)                                 regroup
      = x (x² – 1) + 3 (x² – 1)                              factor each part
      = (x + 3) (x² – 1)                                       a common (x² – 1)
      = (x + 3) (x + 1) (x – 1)

Now that you have the factorization, the polynomial is zero if and only if (at least) one of the factors is zero. (A zero factor makes the function equal to zero, or zero times anything is zero.) So you set each of the (distinct) factors to zero and solve for x:

     x + 3 = 0

x + 3 - 3 = 0 - 3                         subtract 3 from both sides

     x       = -3                             and simplify

Similarly,

x + 1 = 0 gives x = -1 and

x - 1 = 0 gives x = 1

So the zeros of f(x) are x = -3, -1, and 1, or x ε { -3, -1, 1 }

If you cannot see a common factor, you can use the fact that each zero is of the form ±b/a, where b divides the constant term and a divides the leading coefficient. The possible zeros are ±3 and ±1, so you plug each of these possibles into the function to determine which are true zeros. In this case, x = 3 gives f(x) = 27 + 27 - 3 - 3 = 48, which is not zero. Thus the (true) zeros of the function are -3, -1, and 1. Then each factor is (x - a), for each zero a.

The y-intercept is found by setting x equal to 0:

f(0) = 0 + 0 + 0 - 3    or

f(0) = -3

In summary,

f(x) = x³ + 3x² – x – 3 factors to f(x) = (x + 3) (x + 1) (x – 1).

The zeros are at x = -3, -1, and 1, or x ε { -3, -1, 1 }, which means that (-3, 0), (-1, 0), and (1, 0) are points on the graph of y = f(x). (You can check that these are zeros by plugging them into the function and verifying that f(-3) = f(-1) = f(1) = 0.)

The y-intercept is -3, which means that (0, -3) is one of the points of the graph of y = f(x).