using rational root theorem find all possible zeros of 40x^5 - 42x^4 -107x^3 + 33x - 36

To find all possible rational zeros of the polynomial \( 40x^5 - 42x^4 - 107x^3 + 33x - 36 \) using the **Rational Root Theorem**, follow these steps:

### Rational Root Theorem
The Rational Root Theorem states that any rational root, \( \frac{p}{q} \), of a polynomial is a factor of the constant term divided by a factor of the leading coefficient.

Given the polynomial:

\[
40x^5 - 42x^4 - 107x^3 + 33x - 36
\]

- The constant term \( a_0 = -36 \)
- The leading coefficient \( a_n = 40 \)

According to the Rational Root Theorem, the possible rational roots are of the form \( \frac{p}{q} \), where:
- \( p \) is a factor of the constant term \( -36 \)
- \( q \) is a factor of the leading coefficient \( 40 \)

### Step 1: Find factors of the constant term (-36)

Factors of \( -36 \) are:
\[
\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 9, \pm 12, \pm 18, \pm 36
\]

### Step 2: Find factors of the leading coefficient (40)

Factors of \( 40 \) are:
\[
\pm 1, \pm 2, \pm 4, \pm 5, \pm 8, \pm 10, \pm 20, \pm 40
\]

### Step 3: Form all possible rational roots

The possible rational roots are of the form \( \frac{p}{q} \), where \( p \) is a factor of \( -36 \) and \( q \) is a factor of \( 40 \). This gives the following set of possible rational roots:

\[
\pm 1, \pm 2, \pm 3, \pm 4, \pm 5, \pm 6, \pm 9, \pm 10, \pm 12, \pm 18, \pm 36, \pm \frac{1}{2}, \pm \frac{3}{2}, \pm \frac{5}{2}, \pm \frac{9}{2}, \pm \frac{1}{4}, \pm \frac{3}{4}, \pm \frac{5}{4}, \pm \frac{9}{4}, \pm \frac{1}{5}, \pm \frac{2}{5}, \pm \frac{4}{5}, \pm \frac{6}{5}, \pm \frac{9}{5}, \pm \frac{12}{5}, \pm \frac{1}{8}, \pm \frac{5}{8}, \pm \frac{1}{10}, \pm \frac{3}{10}, \pm \frac{9}{10}, \pm \frac{1}{20}, \pm \frac{9}{20}
\]

### Final List of Possible Rational Zeros:
\[
\pm 1, \pm 2, \pm 3, \pm 4, \pm 5, \pm 6, \pm 9, \pm 10, \pm 12, \pm 18, \pm 36, \pm \frac{1}{2}, \pm \frac{3}{2}, \pm \frac{5}{2}, \pm \frac{9}{2}, \pm \frac{1}{4}, \pm \frac{3}{4}, \pm \frac{5}{4}, \pm \frac{9}{4}, \pm \frac{1}{5}, \pm \frac{2}{5}, \pm \frac{4}{5}, \pm \frac{6}{5}, \pm \frac{9}{5}, \pm \frac{12}{5}, \pm \frac{1}{8}, \pm \frac{5}{8}, \pm \frac{1}{10}, \pm \frac{3}{10}, \pm \frac{9}{10}, \pm \frac{1}{20}, \pm \frac{9}{20}
\]

These are the possible rational zeros of the polynomial \( 40x^5 - 42x^4 - 107x^3 + 33x - 36 \).

Would you like to proceed with checking which of these values are actual zeros?

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### Follow-up questions:
1. How do we check if a given rational root is indeed a zero of a polynomial?
2. Can the Rational Root Theorem be applied to any type of polynomial?
3. What is synthetic division, and how does it help in checking for roots?
4. How do imaginary or complex roots relate to the Rational Root Theorem?
5. What happens if none of the possible rational roots are actual zeros?

### Tip:
When dealing with large numbers of potential rational roots, synthetic division is often the quickest way to verify if a candidate is an actual zero.

Discover how to find all possible rational zeros of the polynomial 40x^5 - 42x^4 - 107x^3 + 33x - 36 using the Rational Root Theorem. This step-by-step guide covers polynomial factorization and key concepts for advanced algebra students. Suitable for Grades 11-12.

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