Write the polynomial as a product of linear factors. $$ x^{4}-x^{3}-5 x^{2}-x-6 $$ $$ (x+[?])(x-3)(x-i)(x+[\square i) $$

Question

UpStudy AI · Accepted Answer

We want to factor

x⁴ – x³ – 5x² – x – 6

into linear factors.

Step 1. Notice that x = 3 is a root (since 3⁴ – 3³ – 5·3² – 3 – 6 = 0), so (x – 3) is a factor.

Step 2. Divide the polynomial by (x – 3). The division gives:

x⁴ – x³ – 5x² – x – 6 = (x – 3)(x³ + 2x² + x + 2)

Step 3. Factor the cubic polynomial by grouping:

x³ + 2x² + x + 2 = (x³ + 2x²) + (x + 2)
             = x²(x + 2) + 1(x + 2)
             = (x² + 1)(x + 2)

Step 4. Notice that x² + 1 factors over the complex numbers as (x – i)(x + i).

Step 5. Therefore, the complete factorization over ℂ is:

(x – 3)(x + 2)(x – i)(x + i)

Since the original problem asked for the factorization in the form (x + [?])(x – 3)(x – i)(x + [□ i]), we see that the missing factors are:

x + [?] is x + 2 and x + [□ i] is x + i.

Thus, the answer is:

(x + 2)(x – 3)(x – i)(x + i)
The polynomial $$ x^{4} - x^{3} - 5x^{2} - x - 6 $$ can be factored as $$ (x + 2)(x - 3)(x - i)(x + i) $$.

Write the polynomial as a product of linear factors. \( x^{4}-x^{3}-5 x^{2}-x-6 \) \( (x+[?])(x-3)(x-i)(x+[\square i) \)