Math Problem Statement

Do 17

Given that x = -6 is a zero of the polynomial x^3 + 2x^2 - 19x + 30, find all remaining zeros of the polynomial.

Solution

Let's solve problem 17 step by step:

The problem states:

"Given that x=6x = -6 is a zero of the polynomial x3+2x219x+30x^3 + 2x^2 - 19x + 30, find all remaining zeros of the polynomial."


Step 1: Use Synthetic Division

We divide x3+2x219x+30x^3 + 2x^2 - 19x + 30 by x+6x + 6 (since x=6x = -6 is a root).

Set up synthetic division:

-6 & 1 & 2 & -19 & 30 \\ & & -6 & 24 & -30 \\ \hline & 1 & -4 & 5 & 0 \\ \end{array}$$ The quotient is $$x^2 - 4x + 5$$, and the remainder is 0 (as expected since $$x = -6$$ is a zero). --- ### Step 2: Solve the Quadratic $$x^2 - 4x + 5 = 0$$ To find the remaining zeros, solve the quadratic: $$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)}$$ $$x = \frac{4 \pm \sqrt{16 - 20}}{2}$$ $$x = \frac{4 \pm \sqrt{-4}}{2}$$ $$x = \frac{4 \pm 2i}{2}$$ $$x = 2 \pm i$$ --- ### Step 3: Final Answer The zeros of the polynomial are: $$x = -6, \quad x = 2 + i, \quad x = 2 - i$$ Would you like me to clarify any step or explore further details? --- ### Follow-up Questions: 1. Do you want me to verify this result graphically or with another method? 2. Do you need a deeper explanation of synthetic division? 3. Should I explain the quadratic formula in more detail? 4. Do you want to analyze how complex roots occur in polynomials? 5. Would you like me to expand on how this method works for higher-degree polynomials? --- ### Tip: Remember, complex roots always occur in conjugate pairs for polynomials with real coefficients!

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Math Problem Analysis

Mathematical Concepts

Synthetic Division
Quadratic Equations
Complex Numbers

Formulas

Quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a

Theorems

Fundamental Theorem of Algebra

Suitable Grade Level

Grades 9-12