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)=x^3+4x^2−3x−6

Verified step by step guidance

1. Step 1: Identify the possible rational zeros using the Rational Root Theorem. The possible rational zeros are the factors of the constant term divided by the factors of the leading coefficient. For the polynomial \( f(x) = x^3 + 4x^2 - 3x - 6 \), the constant term is -6 and the leading coefficient is 1. Therefore, the possible rational zeros are \( \pm 1, \pm 2, \pm 3, \pm 6 \).
2. Step 2: Use synthetic division to test each possible rational zero. Start with \( x = 1 \) and perform synthetic division with the coefficients \( 1, 4, -3, -6 \).
3. Step 3: If the remainder is zero, then the tested value is a zero of the polynomial. If not, continue testing the next possible rational zero using synthetic division.
4. Step 4: Once an actual zero is found, use the quotient from the synthetic division to write the polynomial as a product of \( (x - \text{zero}) \) and the quotient polynomial.
5. Step 5: Solve the quotient polynomial to find the remaining zeros. If the quotient is a quadratic, use the quadratic formula or factorization to find the remaining zeros.

Key Concepts

Rational Root Theorem

The Rational Root Theorem provides a method for identifying all possible rational zeros of a polynomial function. It states that any rational solution, expressed as a fraction p/q, must have p as a factor of the constant term and q as a factor of the leading coefficient. This theorem is essential for narrowing down potential candidates for zeros before testing them.

Synthetic Division

Synthetic division is a simplified form of polynomial long division that allows for efficient division of a polynomial by a linear factor. It is particularly useful for testing potential zeros identified through the Rational Root Theorem. By performing synthetic division, one can determine if a candidate zero is indeed a root and obtain the quotient polynomial for further analysis.

Finding Remaining Zeros

Once an actual zero is found using synthetic division, the quotient polynomial can be analyzed to find the remaining zeros. This often involves factoring the quotient or applying the quadratic formula if the quotient is a quadratic polynomial. Understanding how to derive and solve for these remaining zeros is crucial for fully solving polynomial equations.