Math Problem Statement

teach me how you factorize this x3−3x2−4x+12

Solution

Let's walk through the process of factorizing the cubic polynomial:

f(x)=x33x24x+12f(x) = x^3 - 3x^2 - 4x + 12

Step 1: Look for possible rational roots

We begin by applying the Rational Root Theorem, which suggests that the possible rational roots are factors of the constant term (12) divided by the factors of the leading coefficient (1).

  • Factors of 12: ±1,±2,±3,±4,±6,±12\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12
  • Factors of 1 (leading coefficient): ±1\pm 1

So, the possible rational roots are: ±1,±2,±3,±4,±6,±12\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12

Step 2: Test possible roots by substituting into the polynomial

We test these values by substituting them into f(x)f(x). Let's start by testing x=2x = 2:

f(2)=233(22)4(2)+12f(2) = 2^3 - 3(2^2) - 4(2) + 12 f(2)=8128+12=0f(2) = 8 - 12 - 8 + 12 = 0

Since f(2)=0f(2) = 0, x=2x = 2 is a root of the polynomial.

Step 3: Perform synthetic division

Now that we know x=2x = 2 is a root, we can divide the polynomial f(x)f(x) by (x2)(x - 2) using synthetic division.

Set up the synthetic division:

2 & 1 & -3 & -4 & 12 \\ & & 2 & -2 & -12 \\ \hline & 1 & -1 & -6 & 0 \\ \end{array}$$ The result of the division is: $$f(x) = (x - 2)(x^2 - x - 6)$$ ### Step 4: Factor the quadratic Now, we factor the quadratic $$x^2 - x - 6$$. We need to find two numbers that multiply to $$-6$$ and add up to $$-1$$. These numbers are $$-3$$ and $$2$$. Thus, we can factor $$x^2 - x - 6$$ as: $$x^2 - x - 6 = (x - 3)(x + 2)$$ ### Step 5: Final factorization Putting it all together, the complete factorization of $$x^3 - 3x^2 - 4x + 12$$ is: $$f(x) = (x - 2)(x - 3)(x + 2)$$ ### Summary of steps: 1. Apply the Rational Root Theorem to find possible rational roots. 2. Use synthetic division to divide the polynomial by $$(x - 2)$$. 3. Factor the resulting quadratic expression. 4. Combine the factors to get the final factorization. Would you like more details on any of these steps, or do you have any questions? Here are some related topics: 1. How to use the Rational Root Theorem effectively. 2. Steps for synthetic division. 3. Understanding the factorization of quadratics. 4. How to check for other possible roots. 5. Why synthetic division works for dividing polynomials. **Tip:** Always start factoring by checking if the polynomial has simple rational roots first! It can save you a lot of work.

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Math Problem Analysis

Mathematical Concepts

Algebra
Polynomial Factorization
Cubic Equations
Synthetic Division
Quadratic Equations

Formulas

Rational Root Theorem
Synthetic Division
Factoring Quadratics

Theorems

Rational Root Theorem

Suitable Grade Level

Grades 9-11