Review:

We started with quadratic functions (2nd degree polynomials)

Then we talked about polynomial functions . Defined roots /zeros.

If r is a real zero , then what do we know is a factor of the polynomial .

How does synthetic division relate to this?

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Now we will start putting all of these ideas together to find real zeros of a

polynomial function, not given to us in factored form.

Factor Theorem: Let f be a polynomial function. Then (x-c)
is a factor of f(x) if

and only if f(c) = 0.

Determine whether x-c is a factor of f. If it is, write f
in factored form.

f (x) = 3x^{6} + 2x^{3} −176 ; c = -2

Now starting with a polynomial. How can we get the first
zero?

Hopefully there is a zero that is not only real, but also
rational.

Rational Zeros Theorem: Suppose we have a polynomial
function and each

coefficient is an integer . Then if there is a rational zero, p/q, then p is a
factor of

the last coefficient and q must be a factor of the first coefficient.

ex.f (x) = 2x^{3} +11x^{2} − 7s −6 So we can list all of the
possible rational zeros

of f(x). They are all the ( positive and negative ) factors of 6, divided by all
of the

factors of 2. Now we can look at the graph to see which one may work.

Once this is done we can use synthetic division to factor
the polynomial. Why?

Let us review all of these steps .

We are given a polynomial function and want to find the real zeros.

1. List all of the possible rational zeros.

2. Use the graph to pick one that works.

3. Use synthetic division to factor the polynomial.

4. Now what you see is still the same function, only it has been factored. There

is the linear factor (x-c) where c is your rational zero. The other part is the

quotient from the synthetic division. It is called the **depressed equation.**

5. To continue finding real zeros of f, we now try to find real zeros of the

depressed equation. The process is long and yes, depressing.

6. But, as the depressed equation is always a lower degree polynomial, we may

eventually be able to factor it by easier methods like the quadratic formula.

Also, it may not be able to be factored at all and we are done finding real
zeros.

Examples:

f (x) = 2x^{3} −13x^{2} + 24x − 9

f (x) = 2x^{4} +17x ^{3} + 35x^{2} −9x − 45

f (x) = 4 x^{5} −8x^{4} − x + 2