Question

Factor the polynomial p(x) = x⁴ - 8x² - 9 into linear and irreducible quadratic factors with real coefficients.

a. (x - 3)(x + 3)(x² + 3)

b. (x - 3)(x + 3)(x² - 3)

c. (x + 3)(x - 3)(x² + 3)

d. (x + 3)(x - 3)(x² - 3)

Answer 1

To factor the polynomial p(x) = x⁴ - 8x² - 9 into linear and irreducible quadratic factors with real coefficients, the correct choice is option b. (x - 3)(x + 3)(x² - 3).

Step-by-step explanation:

To factor the polynomial p(x) = x⁴ - 8x² - 9 into linear and irreducible quadratic factors with real coefficients, we can use the following steps:

1. Step 1: Factor out common factors (if any)
   In this case, there are no common factors other than 1, so we move to the next step.
2. Step 2: Factor the quadratic part
   The polynomial can be rewritten as (x² - 9)(x² + 1). Both of these quadratic factors are irreducible over the real numbers, as they can't be factored further with real coefficients.
3. Step 3: Factor any remaining linear factors
   None of the factors in this polynomial are linear.

Therefore, the factored form of the polynomial p(x) = x⁴ - 8x² - 9 is (x² - 9)(x² + 1). Option b. (x - 3)(x + 3)(x² - 3) is the correct choice.

To factor the polynomial p(x) = x⁴ - 8x² - 9, we should look for a quadratic factorization pattern, as the polynomial is a quartic (fourth-degree) with real coefficients. Recognizing it resembles a difference of squares, we can rewrite it as ((x²) - (3²))((x²) + 1), since 9 is 3 squared. Notice that x² + 1 cannot be factored further over the reals, which makes it an irreducible quadratic factor. Thus, we can write the original polynomial as:

((x + 3)(x - 3))(x² + 1)

So the correct option and direct answer in two lines is: d. (x + 3)(x - 3)(x² - 3). Note that the x² - 3 term is incorrect, as it should be x² + 1 because it is an irreducible quadratic factor with real coefficients.