For each polynomial function, one zero is given. Find all other zeros. $\text{ƒ}\left(x\right)=x^3-x^2-4x-6;3$

Verified step by step guidance

1. Identify the given polynomial function: \(f(x) = x^3 - x^2 - 4x - 6\), and the given zero: \(x = 3\).
2. Use the fact that if \(x = 3\) is a zero, then \((x - 3)\) is a factor of the polynomial. Perform polynomial division or synthetic division to divide \(f(x)\) by \((x - 3)\).
3. Set up synthetic division with 3 as the divisor and the coefficients of \(f(x)\): 1 (for \(x^3\)), -1 (for \(x^2\)), -4 (for \(x\)), and -6 (constant term).
4. Carry out the synthetic division to find the quotient polynomial, which will be a quadratic polynomial.
5. Solve the quadratic quotient polynomial using factoring, completing the square, or the quadratic formula to find the remaining zeros of \(f(x)\).

Key Concepts

Polynomial Zeros and Roots

Zeros of a polynomial are the values of x that make the function equal to zero. Finding all zeros involves solving the polynomial equation, which can include real and complex roots. Knowing one zero helps in factoring the polynomial to find the remaining zeros.

Polynomial Division (Synthetic or Long Division)

Polynomial division is used to divide the original polynomial by a factor corresponding to a known zero. Synthetic division is a shortcut method for dividing by linear factors, simplifying the polynomial to a lower degree, which makes finding other zeros easier.

Factoring and Solving Quadratic Equations

After dividing the polynomial, the quotient is often a quadratic expression. Factoring or using the quadratic formula on this expression helps find the remaining zeros. This step is essential to completely solve for all roots of the original polynomial.