How does synthetic division relate to this?
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.
De termine whether x-c is a factor of f. If it is, write f
in factored form.
f (x) = 3x6 + 2x3 −176 ; c = -2
Now starting with a polynomial. How can we get the first
Hopefully there is a zero that is not only real, but also
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
the last coefficient and q must be a factor of the first coefficient.
ex.f (x) = 2x3 +11x2 − 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
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?
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 de pressed 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.
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