Math Problem Statement

Write the polynomial as the product of linear factors and list all the zeros of the function. f(x) = x^4 - 4x^3 + 8x^2 - 16x - 16

Solution

The given polynomial is:

f(x)=x44x3+8x216x16f(x) = x^4 - 4x^3 + 8x^2 - 16x - 16

Step 1: Try factoring by grouping or synthetic division

We can attempt to find the factors of the polynomial by checking possible rational roots using the Rational Root Theorem. This theorem states that any rational root of a polynomial with integer coefficients must be a factor of the constant term (in this case, 16-16) divided by a factor of the leading coefficient (which is 11).

The potential rational roots are the factors of 16-16, which are: ±1,±2,±4,±8,±16\pm 1, \pm 2, \pm 4, \pm 8, \pm 16

Let's try synthetic division to check if any of these values are roots.

Step 2: Synthetic division

Let's test x=2x = 2 using synthetic division.

2 & 1 & -4 & 8 & -16 & -16 \\ & & 2 & -4 & 8 & -16 \\ \hline & 1 & -2 & 4 & -8 & 0 \end{array}$$ The remainder is $$0$$, so $$x = 2$$ is a root of the polynomial. Therefore, $$f(x)$$ can be factored as: $$f(x) = (x - 2)(x^3 - 2x^2 + 4x - 8)$$ ### Step 3: Factor the cubic polynomial Next, we factor $$x^3 - 2x^2 + 4x - 8$$. Again, we can try synthetic division or factor by grouping. Let's attempt synthetic division with $$x = 2$$ again. $$\begin{array}{r|rrrr} 2 & 1 & -2 & 4 & -8 \\ & & 2 & 0 & 8 \\ \hline & 1 & 0 & 4 & 0 \end{array}$$ Since the remainder is $$0$$, $$x = 2$$ is also a root of the cubic. Therefore, we have: $$x^3 - 2x^2 + 4x - 8 = (x - 2)(x^2 + 4)$$ ### Step 4: Final factorization Now, we combine all the factors: $$f(x) = (x - 2)^2(x^2 + 4)$$ ### Step 5: Find the zeros To find the zeros of the polynomial, set each factor equal to zero: 1. $$x - 2 = 0 \Rightarrow x = 2$$ 2. $$x^2 + 4 = 0 \Rightarrow x^2 = -4 \Rightarrow x = \pm 2i$$ Thus, the zeros of the function are: $$x = 2 \quad (\text{with multiplicity 2}), \quad x = 2i, \quad x = -2i$$ ### Final Answer: The polynomial written as the product of linear factors is: $$f(x) = (x - 2)^2(x - 2i)(x + 2i)$$ The zeros of the function are: $$x = 2 \ (\text{twice}), \quad x = 2i, \quad x = -2i$$ Would you like further details on any of the steps or clarifications? --- ### Related Questions: 1. How do you perform synthetic division for polynomials? 2. What is the Rational Root Theorem, and how is it used? 3. How do you factor complex polynomials like $$x^2 + 4$$? 4. Can all polynomials be factored into linear factors? 5. How does the multiplicity of a root affect the graph of a polynomial? **Tip:** When factoring polynomials, always check for possible rational roots using the Rational Root Theorem to simplify the factorization process.

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Math Problem Analysis

Mathematical Concepts

Polynomial Factorization
Zeros of Polynomials
Complex Numbers
Synthetic Division

Formulas

Factorization of polynomials
Quadratic equations: x^2 + 4 = 0
Complex roots: x = ±2i

Theorems

Rational Root Theorem

Suitable Grade Level

Grades 10-12