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.)
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Solution

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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: \[ f(x) = (x - 11)(x + 1)(x + 2) \]

Final Answer

The real zeros of \( f \) are: \[ \boxed{x = 11, -1, -2} \]

The factored form of \( f(x) \) is: \[ \boxed{f(x) = (x - 11)(x + 1)(x + 2)} \]

The answer is A.

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