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 f(x)=x36x2+13x20f(x) = x^3 - 6x^2 + 13x - 20, we solve the equation f(x)=0f(x) = 0.

Step 1: Try possible rational roots

Using the Rational Root Theorem, the possible rational roots are factors of the constant term (20-20) divided by factors of the leading coefficient (11):

±1,±2,±4,±5,±10,±20\pm 1, \pm 2, \pm 4, \pm 5, \pm 10, \pm 20

Step 2: Test possible roots

We substitute these values into f(x)f(x) to find a root.

  1. Test x=2x = 2: f(2)=(2)36(2)2+13(2)20=824+2620=0f(2) = (2)^3 - 6(2)^2 + 13(2) - 20 = 8 - 24 + 26 - 20 = 0 Thus, x=2x = 2 is a root.

Step 3: Perform synthetic division

Divide f(x)f(x) by (x2)(x - 2) 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

By Image

Drop file here or Click Here to upload

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