In Exercises 9–16, 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)=x³+x²−4x−4

Verified step by step guidance

1. Identify the polynomial function: $f(x)=x^3+x^2-4x-4$.
2. List all possible rational zeros using the Rational Root Theorem. The possible rational zeros are of the form $\backslash pm\; \backslash frac\{p\}\{q\}$, where $p$ divides the constant term (-4) and $q$ divides the leading coefficient (1). So, possible rational zeros are $\backslash pm\; 1,\; \backslash pm\; 2,\; \backslash pm\; 4$.
3. Use synthetic division to test each possible rational zero by dividing the polynomial by $x\; -\; r$, where $r$ is a candidate zero. Perform synthetic division with each candidate until you find one that gives a remainder of zero, indicating an actual zero of the polynomial.
4. Once an actual zero $r$ is found, write the quotient polynomial from the synthetic division. This quotient will be a quadratic polynomial since the original was cubic.
5. Solve the quadratic quotient polynomial using factoring, completing the square, or the quadratic formula to find the remaining zeros of the original polynomial function.

Key Concepts

Rational Root Theorem

The Rational Root Theorem helps identify all possible rational zeros of a polynomial by considering factors of the constant term and the leading coefficient. These possible roots are expressed as ±(factors of constant term)/(factors of leading coefficient), providing a finite list to test.

Synthetic Division

Synthetic division is a streamlined method for dividing a polynomial by a binomial of the form (x - c). It efficiently tests whether a candidate root is an actual zero by checking if the remainder is zero, simplifying the polynomial to a lower degree for further analysis.

Finding Remaining Zeros Using the Quotient Polynomial

Once a zero is found and the polynomial is divided, the quotient polynomial has a reduced degree. Solving this simpler polynomial (by factoring or using the quadratic formula) reveals the remaining zeros, completing the factorization of the original polynomial.