Math Problem Statement
Find all zeros of f(x)=x3-6x2+13x-20. Enter the zeros separated by commas.
Solution
To find the zeros of , we solve the equation .
Step 1: Try possible rational roots
Using the Rational Root Theorem, the possible rational roots are factors of the constant term () divided by factors of the leading coefficient ():
Step 2: Test possible roots
We substitute these values into to find a root.
- Test : Thus, is a root.
Step 3: Perform synthetic division
Divide by using synthetic division:
2 & 1 & -6 & 13 & -20 \\ & & 2 & -8 & 10 \\ \hline & 1 & -4 & 5 & 0 \\ \end{array}$$ The quotient is $$x^2 - 4x + 5$$, so: $$f(x) = (x - 2)(x^2 - 4x + 5)$$ ### Step 4: Solve the quadratic $$x^2 - 4x + 5 = 0$$ Use the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ Here, $$a = 1$$, $$b = -4$$, $$c = 5$$: $$x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(1)(5)}}{2(1)} = \frac{4 \pm \sqrt{16 - 20}}{2} = \frac{4 \pm \sqrt{-4}}{2}$$ $$x = \frac{4 \pm 2i}{2} = 2 \pm i$$ ### Step 5: List all roots The zeros of $$f(x)$$ are: $$\boxed{2, 2 + i, 2 - i}$$ Would you like a detailed explanation of any step? ### Related Questions: 1. How does the Rational Root Theorem work for higher-degree polynomials? 2. What are the properties of complex roots of polynomials with real coefficients? 3. How can synthetic division simplify solving cubic equations? 4. How would the graph of $$f(x) = x^3 - 6x^2 + 13x - 20$$ look, given these roots? 5. Can $$f(x)$$ have any additional roots if we extend to higher-dimensional numbers? ### Tip: Always verify your roots by substituting them back into the original polynomial!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Algebra
Polynomial Functions
Complex Numbers
Formulas
Rational Root Theorem
Quadratic formula: x = (-b ± √(b² - 4ac)) / 2a
Synthetic division for polynomial division
Theorems
Rational Root Theorem
Fundamental Theorem of Algebra
Suitable Grade Level
Grades 9-12
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