Math Problem Statement
Sistem persamaan polinomial adalah sistem persamaan dengan pangkat tertingginya lebih besar dari 2 ( > 2). Dengan menggunakan konsep persamaan polinomial, tentukan faktor-faktor linear dari polinomial x^3 - 7x + 6.
Solution
The problem is asking for the linear factors of the polynomial using polynomial concepts.
Step 1: Apply the Rational Root Theorem
To find potential roots, consider the factors of the constant term (6) and the leading coefficient (1). The possible rational roots are:
Step 2: Test for Roots
Substitute the potential roots into .
- For : is a root.
Step 3: Perform Synthetic Division
Divide by .
1 & 1 & 0 & -7 & 6 \\ & & 1 & 1 & -6 \\ \hline & 1 & 1 & -6 & 0 \\ \end{array}$$ The quotient is $$x^2 + x - 6$$. ### Step 4: Factorize the Quadratic Polynomial The quadratic $$x^2 + x - 6$$ can be factored as: $$x^2 + x - 6 = (x + 3)(x - 2)$$ ### Step 5: Combine All Factors The complete factorization of $$x^3 - 7x + 6$$ is: $$x^3 - 7x + 6 = (x - 1)(x + 3)(x - 2)$$ ### Final Answer The linear factors of $$x^3 - 7x + 6$$ are: $$(x - 1), (x + 3), (x - 2)$$ Do you want details on any step, or would you like further assistance? ### Related Questions 1. How does the Rational Root Theorem work in polynomial factorization? 2. Can you explain the synthetic division process in detail? 3. How can we verify that the factors are correct? 4. What other methods can be used to factorize higher-degree polynomials? 5. How do linear factors relate to the graph of the polynomial? ### Tip Always check all possible roots from the Rational Root Theorem to ensure all linear factors are accounted for.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Polynomial Equations
Rational Root Theorem
Synthetic Division
Factoring
Formulas
f(x) = x^3 - 7x + 6
Factoring x^2 + bx + c
Theorems
Rational Root Theorem
Suitable Grade Level
Grades 10-12