Questions: Find the zeros and fully factor (f(x)=x^3-2 x^2-7 x+2), including factors for irrational zeros. Use radicals, not decimal approximations. The zeros are The fully factored form is (f(x)=)

Find the zeros and fully factor (f(x)=x^3-2 x^2-7 x+2), including factors for irrational zeros. Use radicals, not decimal approximations.

The zeros are
The fully factored form is (f(x)=)

Transcript text: Find the zeros and fully factor $f(x)=x^{3}-2 x^{2}-7 x+2$, including factors for irrational zeros. Use radicals, not decimal approximations. The zeros are $\square$ The fully factored form is $f(x)=$ $\square$

Solution

Solution Steps

Step 1: Finding the Zeros

To find the zeros of the polynomial $ f(x) = x^3 - 2x^2 - 7x + 2 $, we first identify the rational roots. The rational roots are found to be:

\[ x = -2, \quad x = 2 - \sqrt{3}, \quad x = 2 + \sqrt{3} \]

Step 2: Factoring the Polynomial

Next, we factor the polynomial using the identified rational root $ x = -2 $. Performing synthetic division gives us:

\[ f(x) = (x + 2)(x^2 - 4x + 1) \]

Step 3: Solving the Quadratic Equation

We then solve the quadratic equation $ x^2 - 4x + 1 = 0 $ using the quadratic formula:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{4 \pm \sqrt{16 - 4}}{2} = \frac{4 \pm \sqrt{12}}{2} = 2 \pm \sqrt{3} \]

Thus, the zeros of the polynomial are confirmed as:

\[ x = -2, \quad x = 2 - \sqrt{3}, \quad x = 2 + \sqrt{3} \]

Final Answer

The zeros are:

\[ \boxed{-2, \; 2 - \sqrt{3}, \; 2 + \sqrt{3}} \]

The fully factored form of the polynomial is:

\[ \boxed{f(x) = (x + 2)(x^2 - 4x + 1)} \]

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