Math Problem Statement

Find all the zeros of f(x) = x^3 + 2x^2 - 15x - 36.

Solution

The given function is:

f(x)=x3+2x2−15x−36f(x) = x^3 + 2x^2 - 15x - 36

It appears that x=−3x = -3 is one of the roots (since it is already filled in). To find the remaining roots, we can use synthetic division or factor the cubic equation.

Step 1: Synthetic Division

We'll perform synthetic division with x=−3x = -3 to reduce the cubic polynomial to a quadratic one.

The coefficients of f(x)f(x) are: 1,2,−15,−361, 2, -15, -36.

Using synthetic division:

-3 & 1 & 2 & -15 & -36 \\ & & -3 & 3 & 36 \\ \hline & 1 & -1 & -12 & 0 \\ \end{array}$$ The quotient is $$x^2 - x - 12$$. ### Step 2: Factor the Quadratic Now, factor the quadratic: $$x^2 - x - 12 = (x - 4)(x + 3)$$ So, the remaining roots are $$x = 4$$ and $$x = -3$$. ### Step 3: List All the Roots Since $$x = -3$$ is a double root, we list it twice. Thus, the zeros of the function are: $$x = -3, -3, 4$$ Would you like further details on this process or have any questions? Here are five related questions: 1. How is synthetic division used to find roots of polynomials? 2. What is the difference between simple and double roots? 3. How can you verify if a root is correct for a given polynomial? 4. What are the other methods for solving cubic equations besides factoring? 5. How does the Factor Theorem relate to solving polynomial equations? **Tip:** When you find a root of a cubic equation, always divide the cubic by $$x - \text{root}$$ to reduce the degree and make further solving easier.

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Math Problem Analysis

Mathematical Concepts

Algebra
Polynomials
Synthetic Division
Factoring

Formulas

Cubic Equation: ax^3 + bx^2 + cx + d = 0
Quadratic Formula
Factoring

Theorems

Factor Theorem
Synthetic Division

Suitable Grade Level

Grades 10-12

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