Questions: Question 5 of 24 This test 24 point(s) possible This question: 1 point(s) possible Submit 1 Use the rational zeros theorem to find all the real zeros of the polynomial function. Use the zeros to factor f over the real numbers. f(x)=x^3-8x^2-31x-22 Find the real zeros of f. Select the correct choice below and, if necessary, fill in the answer box to complete your answer. A. x= (Simplify your answer. Type an exact answer, using radicals as needed. Use integers or fractions for any rational numbers in the expression. Use a comma separate answers as needed.) B. There are no real zeros. Use the real zeros to factor f. f(x)= (Simplify your answer. Type your answer in factored form. Type an exact answer, using radicals as needed. Use integers or fractions for any rational numbers in the expression.)

$Question 5 of 24 This test 24 point(s) possible This question: 1 point(s) possible Submit 1 Use the rational zeros theorem to find all the real zeros of the polynomial function. Use the zeros to factor f over the real numbers. f(x)=x^3-8x^2-31x-22 Find the real zeros of f. Select the correct choice below and, if necessary, fill in the answer box to complete your answer. A. x= (Simplify your answer. Type an exact answer, using radicals as needed. Use integers or fractions for any rational numbers in the expression. Use a comma separate answers as needed.) B. There are no real zeros. Use the real zeros to factor f. f(x)= (Simplify your answer. Type your answer in factored form. Type an exact answer, using radicals as needed. Use integers or fractions for any rational numbers in the expression.)$

Transcript text: Question 5 of 24 This test 24 point(s) possible This question: 1 point(s) possible Submit 1 Use the rational zeros theorem to find all the real zeros of the polynomial function. Use the zeros to factor $f$ over the real numbers. \[ f(x)=x^{3}-8 x^{2}-31 x-22 \] Find the real zeros of $f$. Select the correct choice below and, if necessary, fill in the answer box to complete your answer. A. $x=$ $\square$ (Simplify your answer. Type an exact answer, using radicals as needed. Use integers or fractions for any rational numbers in the expression. Use a comma separate answers as needed.) B. There are no real zeros. Use the real zeros to factor $f$. \[ f(x)= \] $\square$ (Simplify your answer. Type your answer in factored form. Type an exact answer, using radicals as needed. Use integers or fractions for any rational numbers in the expression.)

Solution

Solution Steps

Step 1: Apply the Rational Root Theorem

To find the real zeros of the polynomial $ f(x) = x^3 - 8x^2 - 31x - 22 $, we first list all possible rational zeros using the Rational Root Theorem. The possible rational zeros are the factors of the constant term (-22) divided by the factors of the leading coefficient (1).

Step 2: Test Possible Rational Zeros

We test each possible rational zero by substituting it into the polynomial to find the actual zeros. The actual zeros are: \[ x = 11, \quad x = -1, \quad x = -2 \]

Step 3: Factor the Polynomial

Using the real zeros, we can factor the polynomial $ f(x) $ into a product of binomials: \[ f(x) = (x - 11)(x + 1)(x + 2) \]