For Exercises 40–46, (a) List all possible rational roots or rational zeros. (b) Use Descartes's Rule of Signs to determine the possible number of positive and negative real roots or real zeros. (c) Use synthetic division to test the possible rational roots or zeros and find an actual root or zero. (d) Use the quotient from part (c) to find all the remaining roots or zeros. $f\left(x\right)=x^3+3x^2-4$

Verified step by step guidance

1. For part (a), to list all possible rational roots of the polynomial \(f(x) = x^3 + 3x^2 - 4\), use the Rational Root Theorem. Identify the factors of the constant term and the leading coefficient. The constant term is \(-4\), and the leading coefficient is \(1\). List all possible rational roots as \(\pm \frac{p}{q}\), where \(p\) divides the constant term and \(q\) divides the leading coefficient.
2. For part (b), apply Descartes's Rule of Signs to determine the possible number of positive real roots by counting the number of sign changes in \(f(x) = x^3 + 3x^2 - 4\). Then, find the possible number of negative real roots by evaluating \(f(-x)\) and counting the sign changes in that polynomial.
3. For part (c), use synthetic division to test each possible rational root found in part (a). Set up synthetic division with the root candidate and the coefficients of \(f(x)\), which are \(1\) (for \(x^3\)), \(3\) (for \(x^2\)), \(0\) (for \(x\)), and \(-4\) (constant term). Perform the synthetic division to check if the remainder is zero, indicating a root.
4. For part (d), once you find a root from part (c), use the quotient polynomial obtained from synthetic division. This quotient will be a quadratic polynomial. Solve this quadratic polynomial using factoring, completing the square, or the quadratic formula to find the remaining roots.
5. Summarize all roots found: the root from synthetic division and the roots from solving the quadratic quotient. These roots together represent all the zeros of the original cubic polynomial.

Key Concepts

Rational Root Theorem

The Rational Root Theorem helps identify all possible rational roots 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). This theorem narrows down candidates for testing actual roots.

Descartes's Rule of Signs

Descartes's Rule of Signs estimates the number of positive and negative real roots of a polynomial by counting sign changes in the coefficients. The number of positive roots equals the number of sign changes or less by an even number; similarly, for negative roots, apply the rule to f(-x). This guides expectations about root distribution.

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 c is an actual root by checking if the remainder is zero. It also produces a quotient polynomial used to find remaining roots.