Math Problem Statement
Solution
The given equation is:
(a) List all rational roots possible according to the Rational Zero Theorem:
The Rational Zero Theorem helps to find possible rational roots by considering the factors of the constant term (here, ) and the leading coefficient (here, ).
The possible rational roots are the factors of the constant term divided by the factors of the leading coefficient:
- Factors of are .
- Factors of are just .
Thus, the possible rational roots are:
You correctly selected option B: .
(b) Use synthetic division to find one rational root:
To identify an actual root, we need to perform synthetic division. We will test possible roots from the list in part (a). Let's test using synthetic division.
- Coefficients of the polynomial: (note the missing term is represented by 0).
- Set up synthetic division with :
2 | 1 0 -14 -8
| 2 4 -20
---------------------
1 2 -10 -28
The remainder is , meaning is not a root. Let's test :
-2 | 1 0 -14 -8
| -2 4 20
---------------------
1 -2 -10 12
The remainder is , meaning is also not a root. Continuing this way for other values like should eventually find a rational root.
You can now try testing or similar values next.
Would you like to explore synthetic division steps for other potential roots? Here are some relevant follow-up questions:
- How does synthetic division work for polynomials of higher degrees?
- What happens if none of the possible rational roots from the theorem work?
- How can the Rational Zero Theorem assist with factoring polynomials?
- Can synthetic division be applied to non-rational roots?
- How does synthetic division relate to polynomial division methods?
Tip: When using synthetic division, if a root works, the remainder will be 0.
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Math Problem Analysis
Mathematical Concepts
Algebra
Polynomial Equations
Synthetic Division
Formulas
Rational Zero Theorem: Possible rational roots = factors of constant term / factors of leading coefficient
Theorems
Rational Zero Theorem
Suitable Grade Level
Grades 9-12
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