Cubic Polynomial Zeros and Factored Form: x^3 + 4x^2 + 9x + 36

Math Problem Statement

Find all the zeros (real and complex) of P(x) = x^3 + 4x^2 + 9x + 36. Write P in factored form as a product of linear and irreducible quadratic factors.

Solution

Let's analyze the given polynomial $P(x) = x^3 + 4x^2 + 9x + 36$ and find the zeros and its factored form.

Step 1: Finding the zeros of $P(x)$

We begin by attempting to find rational roots using the Rational Root Theorem. The possible rational roots are the divisors of the constant term (36) divided by the divisors of the leading coefficient (1). So, the possible rational roots are: $\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 9, \pm 12, \pm 18, \pm 36$

Testing $x = -3$ using synthetic division:

Synthetic Division for $x = -3$ :

Coefficients of $P(x)$ : $1, 4, 9, 36$
Perform synthetic division:

-3 & 1 & 4 & 9 & 36 \\ & & -3 & -3 & -18 \\ \hline & 1 & 1 & 6 & 18 \\ \end{array}$$ The remainder is zero, so $$x = -3$$ is a root. This means $$P(x) = (x + 3)(x^2 + x + 12)$$. ### Step 2: Factor the quadratic $$x^2 + x + 12$$ To factor $$x^2 + x + 12$$, we need to find its roots using the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ For $$x^2 + x + 12$$, $$a = 1$$, $$b = 1$$, and $$c = 12$$: $$x = \frac{-1 \pm \sqrt{1^2 - 4(1)(12)}}{2(1)} = \frac{-1 \pm \sqrt{1 - 48}}{2} = \frac{-1 \pm \sqrt{-47}}{2}$$ $$x = \frac{-1 \pm \sqrt{47}i}{2}$$ Thus, the two complex roots are $$x = \frac{-1 + \sqrt{47}i}{2}$$ and $$x = \frac{-1 - \sqrt{47}i}{2}$$. ### Step 3: Final factored form The full factored form of $$P(x)$$ is: $$P(x) = (x + 3) \left( x^2 + x + 12 \right)$$ ### Answer: - **Zeros**: $$-3, \frac{-1 + \sqrt{47}i}{2}, \frac{-1 - \sqrt{47}i}{2}$$ - **Factored form**: $$P(x) = (x + 3)(x^2 + x + 12)$$ Would you like more details on any part of this solution? Here are some related questions you might consider: 1. How does the Rational Root Theorem help find potential roots? 2. What are the steps for synthetic division? 3. How does the quadratic formula handle complex roots? 4. Could this polynomial be factored without the quadratic formula? 5. Why is $$\sqrt{47}i$$ involved in complex roots? **Tip**: Always test small integers first when looking for rational roots, as they are often simpler to check.

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Math Problem Analysis

Mathematical Concepts

Polynomials
Complex Numbers
Quadratic Equations
Synthetic Division

Formulas

Quadratic formula: x = (-b ± √(b² - 4ac)) / 2a
Rational Root Theorem: Possible rational roots = ± factors of the constant / factors of the leading coefficient

Theorems

Rational Root Theorem
Quadratic Formula Theorem

Suitable Grade Level

Grades 10-12

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