6.3 Locating real zeros

Rational Zero Theorem: A theorem that helps in identifying all possible rational zeros of a polynomial function.
Definition of Rational Zeros: A rational zero (or a root) of a polynomial function can be expressed as the fraction $\frac{p}{q}$ where:
- p is a factor of the constant term (last term).
- q is a factor of the leading coefficient (first term).
Notation: This theorem provides potential rational zeros but does not guarantee their existence.
- For example, it does not provide information about irrational or complex zeros.

Polynomial Form: Given a polynomial function in standard form:
- $f(x) = ax^n + bx^{n-1} + … + k$ where $a \neq 0$ .
Example to demonstrate:
- For a polynomial $2x^3 + 5x^2 - 4x - 3$ :
- Last term (k): -3
- Leading coefficient (a): 2

Identify Factors of the Last Term (p) and the Leading Coefficient (q):
- Factors of -3: $\pm 1, \pm 3$ .
- Factors of 2: $\pm 1, \pm 2$ .
List Possible Rational Zeros:
- Possible rational zeros come from combinations of factors of p over factors of q:
- $\frac{1}{1}, \frac{1}{-1}, \frac{1}{2}, \frac{1}{-2}, \frac{-1}{1}, \frac{-1}{-1}, \frac{-1}{2}, \frac{-1}{-2}, \frac{3}{1}, \frac{3}{-1}, \frac{3}{2}, \frac{3}{-2}, \frac{-3}{1}, \frac{-3}{-1}, \frac{-3}{2}, \frac{-3}{-2}$ .
- When simplified, the actual potential rational solutions are: $\pm 1, \pm \frac{1}{2}, \pm 3$ .
Using Synthetic Division to Check Rational Zeros:
- Synthetic Division helps determine if a given rational zero is valid.
- If the remainder is zero, then it is a valid zero. If the remainder is not zero, then it's not a valid zero.
- Example: Checking the value 1 (checking work step):
  - Start with 2, then 7, then -3 (not valid, try next).
- Continuing with -1, and further checking zeros reduces options until actual zeros are determined.

Identify actual zeros through confirmed synthetic division results.
After finding a zero, the polynomial reduces to a quadratic or simpler polynomial for further factorization.
Subsequent steps include checking the remaining polynomial for further factors or using quadratic formula if necessary:
- Example finding from the equation: from $0 = 2x^3 + 5x^2 - 4x - 3$ , if one zero found was 1, led to finding a quadratic $(2x^2 + 7x + 3)$ .

Descartes' Rule of Signs provides information about the number of possible positive and negative real roots:
- Count the number of sign changes in the polynomial to determine possible positive roots.
- A polynomial f(x) shows the count of sign shifts gives you number of positive roots.
- Repeating the process with negative x for negative roots.
- Outcomes usually indicate either the exact count or an even deficiency.

Intermediate Value Theorem states:
- If a polynomial function is continuous over an interval and changes signs at two endpoints, then at least one root exists between them.
- Example of testing values within intervals to pinpoint where the crossing occurs may involve verifying the polynomial outputs at given inputs.

Upper and Lower Bound Theorems are used to refine possible values for real zeros by testing values through synthetic division; yielding positive or alternating signs help determine valid ranges for possible zeros.
- If all positive in the last row of synthetic division, leads to upper bound conclusion.
- Alternating signs in the last row suggest a lower bound.

Through systematic applications of the Rational Zero Theorem, synthetic division, Descartes' rule, Intermediate Value Theorem, and bounding techniques, we can effectively identify not just potential but actual zeros of polynomial functions, refining our search for their roots.