Finding Real Zeros and Factoring Polynomial f(x) = x^3 - 10x^2 + 28x

Math Problem Statement

Find the real zeros of f. Use the real zeros to factor f. f(x) = x^3 - 10x^2 + 28x - 16

Solution

To find the real zeros of $f(x) = x^3 - 10x^2 + 28x - 16$ and use them to factor $f(x)$ , let's proceed as follows:

Step 1: Find Possible Rational Zeros

We can apply the Rational Root Theorem, which suggests that any rational root of the polynomial will be a factor of the constant term (-16) divided by a factor of the leading coefficient (1). So, the possible rational roots are: $\pm 1, \pm 2, \pm 4, \pm 8, \pm 16$

Step 2: Test Each Possible Rational Root

We will test these values in $f(x)$ to see if any are actual roots.

Testing $x = 2$ : $f(2) = 2^3 - 10 \cdot 2^2 + 28 \cdot 2 - 16 = 8 - 40 + 56 - 16 = 0$ Since $f(2) = 0$ , $x = 2$ is a root.

Step 3: Use Synthetic Division

Since $x = 2$ is a root, we can divide $f(x)$ by $x - 2$ using synthetic division.

2 & 1 & -10 & 28 & -16 \\ & & 2 & -16 & 24 \\ \hline & 1 & -8 & 12 & 0 \\ \end{array}$$ The quotient is $$x^2 - 8x + 8$$, so we can rewrite $$f(x)$$ as: $$f(x) = (x - 2)(x^2 - 8x + 8)$$ ### Step 4: Factor the Quadratic Now, we need to find the roots of $$x^2 - 8x + 8 = 0$$ using the quadratic formula: $$x = \frac{-(-8) \pm \sqrt{(-8)^2 - 4 \cdot 1 \cdot 8}}{2 \cdot 1} = \frac{8 \pm \sqrt{64 - 32}}{2} = \frac{8 \pm \sqrt{32}}{2}$$ $$x = \frac{8 \pm 4\sqrt{2}}{2} = 4 \pm 2\sqrt{2}$$ So the real zeros of $$f(x)$$ are: $$x = 2, \quad x = 4 + 2\sqrt{2}, \quad x = 4 - 2\sqrt{2}$$ ### Step 5: Write $$f(x)$$ in Factored Form Using the real zeros, we can express $$f(x)$$ as: $$f(x) = (x - 2)\left(x - (4 + 2\sqrt{2})\right)\left(x - (4 - 2\sqrt{2})\right)$$ ### Final Answer The real zeros are $$x = 2$$, $$x = 4 + 2\sqrt{2}$$, and $$x = 4 - 2\sqrt{2}$$, and the factored form of $$f(x)$$ is: $$f(x) = (x - 2)(x - (4 + 2\sqrt{2}))(x - (4 - 2\sqrt{2}))$$ Would you like more details on any of these steps, or do you have questions? ### Related Questions 1. How can you determine if a polynomial has any complex roots? 2. What is the Rational Root Theorem, and how is it applied? 3. How do you perform synthetic division for a polynomial? 4. What steps can be taken if none of the rational roots work? 5. How do you simplify expressions with square roots in factoring? **Tip:** When finding zeros of a polynomial, always start with the Rational Root Theorem to narrow down potential solutions.

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Math Problem Analysis

Mathematical Concepts

Algebra
Polynomial Functions
Roots of Polynomials
Synthetic Division

Formulas

Quadratic formula x = (-b ± √(b^2 - 4ac)) / 2a

Theorems

Rational Root Theorem

Suitable Grade Level

Grades 10-12

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