Math Problem Statement

Find the remaining zeros of f(x) = -x^3 + 5x^2 - 3x - 9, given that c = -1 is a zero. Rewrite f(x) in completely factored form and identify the zeros.

Solution

The given problem involves finding the remaining zeros of the function f(x)=−x3+5x2−3x−9f(x) = -x^3 + 5x^2 - 3x - 9, given that c=−1c = -1 is already a zero. We are also asked to rewrite f(x)f(x) in completely factored form and identify all the zeros.

Step 1: Use synthetic division to find other zeros

Since x=−1x = -1 is a zero, we can use synthetic division to divide f(x)f(x) by (x+1)(x + 1).

The coefficients of the polynomial are: [−1,5,−3,−9][-1, 5, -3, -9] Performing synthetic division with x=−1x = -1:

-1 & -1 & 5 & -3 & -9 \\ & & 1 & -6 & 9 \\ \hline & -1 & 4 & -9 & 0 \\ \end{array}$$ So, the quotient is $$-x^2 + 4x - 9$$. ### Step 2: Solve the quadratic equation We are left with the quadratic $$-x^2 + 4x - 9 = 0$$. Solving for $$x$$ using the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $$a = -1$$, $$b = 4$$, and $$c = -9$$. $$x = \frac{-4 \pm \sqrt{(4)^2 - 4(-1)(-9)}}{2(-1)}$$ $$x = \frac{-4 \pm \sqrt{16 - 36}}{-2}$$ $$x = \frac{-4 \pm \sqrt{-20}}{-2}$$ $$x = \frac{-4 \pm 2i\sqrt{5}}{-2}$$ $$x = 2 \pm i\sqrt{5}$$ Thus, the remaining zeros are $$2 + i\sqrt{5}$$ and $$2 - i\sqrt{5}$$. ### Step 3: Write the completely factored form of $$f(x)$$ The completely factored form of $$f(x)$$ is: $$f(x) = -(x + 1)(x - (2 + i\sqrt{5}))(x - (2 - i\sqrt{5}))$$ Alternatively, since the quadratic can also be factored as $$(x - 2)^2 + 5$$, the factored form of $$f(x)$$ can be written as: $$f(x) = -(x + 1)((x - 2)^2 + 5)$$ Would you like a detailed explanation on any specific part of this solution? Here are 5 related questions you might find useful: 1. How do you perform synthetic division step by step? 2. Can you explain the quadratic formula and its derivation? 3. What are complex roots and how do they appear in factored form? 4. How would you graph a cubic function with complex roots? 5. Why does synthetic division help in finding the roots of a polynomial? **Tip:** When dealing with higher-degree polynomials, always check for rational roots using synthetic or polynomial division to simplify the problem before solving for other roots.

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Math Problem Analysis

Mathematical Concepts

Polynomial Division
Quadratic Formula
Complex Numbers

Formulas

Quadratic Formula: x = (-b ± √(b^2 - 4ac)) / 2a
Synthetic Division for Polynomials

Theorems

Fundamental Theorem of Algebra

Suitable Grade Level

Grades 10-12

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