Math Problem Statement
Solution
To find the possible rational zeros of the cubic polynomial , we can apply the Rational Root Theorem. This theorem states that any possible rational root of a polynomial with integer coefficients will be of the form:
where:
- is a factor of the constant term (in this case, ),
- is a factor of the leading coefficient (in this case, , since the leading term is ).
Step 1: Identify factors of the constant term (8):
The factors of 8 are:
Step 2: Identify factors of the leading coefficient (1):
The only factors of 1 are:
Step 3: List all possible rational zeros:
Since can only be , the possible rational roots are just the factors of 8, which are:
Thus, the possible rational zeros of the polynomial are:
Would you like further details on how to verify if any of these roots are actual solutions? Or any questions about this process?
Related Questions:
- How can we test which of the possible rational roots are actual zeros?
- What is the Rational Root Theorem and how is it used?
- Can we apply synthetic division to verify the roots?
- What would happen if the leading coefficient were not 1?
- How do irrational or complex roots come into play in higher degree polynomials?
Tip: Always double-check potential rational roots by substituting them into the polynomial!
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Math Problem Analysis
Mathematical Concepts
Algebra
Polynomial Equations
Rational Root Theorem
Formulas
Possible rational roots = p/q, where p is a factor of the constant term and q is a factor of the leading coefficient
Theorems
Rational Root Theorem
Suitable Grade Level
Grades 10-12
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