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Chapter 2: Problem 9

a. List all possible rational zeros. b. Use synthetic division to test the possible rational zeros and find an actual zero. c. Use the quotient from part (b) to find the remaining zeros of the polynomial function. $$f(x)=x^{3}+x^{2}-4 x-4$$

Short Answer

Expert verified
The zeros of the polynomial function \(f(x) = x^3 + x^2 - 4x - 4\) are -1 (real) and \( -1 \pm \sqrt{3}i\) (complex).

Step by step solution

01

STEP 1: Identify Rational Zero Candidates

Rational Zero Theorem states that any rational zero of \(f(x) = x^3 + x^2 - 4x - 4\) will be a factor of the constant term (4) divided by a factor of the leading coefficient (1). The factors of 4 are 卤1, 卤2, 卤4, resulting in possible rational zeros of 卤1, 卤2, 卤4.
02

STEP 2: Use Synthetic Division to Find an Actual Zero

Synthetic division can be used to test each potential rational zero. Start with the easier ones, let's take -1. It results in no remainder, making -1 an actual zero of the polynomial.
03

STEP 3: Use the Resulting Quotient to Find the Remaining Zeros

The quotient from the synthetic division in step 2 is \(x^2 + 2x + 4\). Solve \(x^2 + 2x + 4 = 0\) by using the quadratic formula \(\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). This will give the remaining zeros which are complex roots.

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Most popular questions from this chapter

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