Questions: Consider the following polynomial function. f(x) = x^4 - 3x^3 - 17x^2 + 21x + 70 Factor the polynomial completely.

Transcript text: Consider the following polynomial function. \[ f(x)=x^{4}-3 x^{3}-17 x^{2}+21 x+70 \] Factor the polynomial completely.

Solution

Solution Steps

To factor the polynomial completely, we can use numerical methods to find its roots. Once the roots are identified, the polynomial can be expressed as a product of linear factors corresponding to these roots. If any roots are complex, they will appear in conjugate pairs.

Step 1: Identify the Roots

The roots of the polynomial \( f(x) = x^4 - 3x^3 - 17x^2 + 21x + 70 \) are found to be: \[ x_1 = 5, \quad x_2 \approx 2.6458, \quad x_3 \approx -2.6458, \quad x_4 = -2 \]

Step 2: Factor the Polynomial

Using the identified roots, we can express the polynomial in its factored form. The polynomial can be factored as: \[ f(x) = (x - 5)(x - 2.6458)(x + 2.6458)(x + 2) \]

Step 3: Simplify the Factored Form

The factors corresponding to the roots can be grouped as follows: \[ f(x) = (x - 5)\left((x - 2.6458)(x + 2.6458)\right)(x + 2) \] The term \((x - 2.6458)(x + 2.6458)\) can be simplified to: \[ (x^2 - (2.6458)^2) = (x^2 - 7) \] Thus, the complete factored form of the polynomial is: \[ f(x) = (x - 5)(x^2 - 7)(x + 2) \]