Find all the zeros of the function

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Given a function and one of its zeros find all of the zeros of the function:

2x^3 + 7x^2 - 53x - 28; x - 4


Using the rational zero theorem, list all the possible rational zeros for the function:

p(x) = 6x^3 + 4x^2 - 14x + 4

Mar 11th, 2015

Synthethic Division

Since the highest exponential degreee of the equation is (3), then the equation has 3 zeros. 

We already know one of the zeros: (x-4). Thus, we must divide the equation in order to get  new one. We will divide 

(2x^3 + 7x^2 - 53x - 28) / (x-4)    using synthetic division

After divinding, we get a new equation:
2x^2 + 15x + 7
Now, we can use the quadratic formula to find the other 2 zeros.

We know that

a= 2

b= 15


We have found the three zeros:

  1. (x - 4)
  2. (x + 7)
  3. (x + 1/2)

The Rational Zero Theorem. 

You need to list all the possible factors of p (6) and q (4).

Also, all the values of (p/q). [Take into account that there are duplicates, just cross them out]

Now, you must use synthetic division with all values of (p/q) in order to determine if they are zeros. For example, I will use synthethic division with (1/3).

As you can see, there is no remainder. Therefore, 1/3 is one of the equation's zeros. You do the work for the other ones if you need it. 

The three zeros for this equation are:

  1. (x - 1/3)
  2. (x - 1)
  3. (x + 2)

Mar 12th, 2015

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