Math Problem Statement

Use the Factor Theorem and the given binomial to completely factor the polynomial f(x) = x^3 + 6x^2 - 16x - 96; (x - 4) (use Synthetic Division). Then, give the zeros.

Solution

The question asks you to use the Factor Theorem and the given binomial (x−4)(x - 4) to completely factor the polynomial f(x)=x3+6x2−16x−96f(x) = x^3 + 6x^2 - 16x - 96, using synthetic division, and then find the zeros.

Let's break it down step by step:

Step 1: Synthetic Division

We will divide the polynomial f(x)f(x) by x−4x - 4 using synthetic division. First, list the coefficients of the polynomial:

  • Coefficients of f(x)=x3+6x2−16x−96f(x) = x^3 + 6x^2 - 16x - 96 are: [1,6,−16,−96][1, 6, -16, -96].

We now perform synthetic division with the root x=4x = 4 (from x−4=0x - 4 = 0).

Synthetic Division Process:

4 & 1 & 6 & -16 & -96 \\ & & 4 & 40 & 96 \\ \hline & 1 & 10 & 24 & 0 \\ \end{array}$$ - The quotient from the division is $$x^2 + 10x + 24$$ and the remainder is 0, which confirms that $$(x - 4)$$ is indeed a factor. ### Step 2: Factor the Quotient Now, we need to factor the quotient $$x^2 + 10x + 24$$. We'll use the quadratic formula to find its roots: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ For $$x^2 + 10x + 24$$: - $$a = 1$$ - $$b = 10$$ - $$c = 24$$ $$x = \frac{-10 \pm \sqrt{10^2 - 4(1)(24)}}{2(1)} = \frac{-10 \pm \sqrt{100 - 96}}{2} = \frac{-10 \pm \sqrt{4}}{2}$$ $$x = \frac{-10 \pm 2}{2}$$ Thus, the two roots are: $$x = \frac{-10 + 2}{2} = -4 \quad \text{and} \quad x = \frac{-10 - 2}{2} = -6$$ ### Step 3: Write the Complete Factorization Now we can write the complete factorization of $$f(x)$$: $$f(x) = (x - 4)(x + 4)(x + 6)$$ ### Step 4: Find the Zeros The zeros of the polynomial are the values of $$x$$ that make the factors equal to zero: $$x = 4, \quad x = -4, \quad x = -6$$ ### Final Answer: - **Factored form**: $$f(x) = (x - 4)(x + 4)(x + 6)$$ - **Zeros**: $$x = 4, -4, -6$$ Would you like a more detailed explanation of any part of the process? Here are some related questions for practice: 1. How can you verify the factorization of a cubic polynomial? 2. What happens when you divide a polynomial and the remainder is not zero? 3. What is the purpose of the Factor Theorem in factoring polynomials? 4. How does synthetic division simplify the division of polynomials compared to long division? 5. Can a cubic polynomial have more than three real roots? **Tip**: Always double-check your synthetic division by multiplying the quotient back with the divisor to confirm it gives the original polynomial!

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Math Problem Analysis

Mathematical Concepts

Factor Theorem
Synthetic Division
Factoring Polynomials
Zeros of Polynomials

Formulas

Quadratic Formula
Synthetic Division Process

Theorems

Factor Theorem

Suitable Grade Level

Grades 10-12

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