For each polynomial function, one zero is given. Find all other zeros. $\text{ƒ}\left(x\right)=4x^3+6x^2-2x-1;\frac{1}{2}$

Verified step by step guidance

1. Identify the given polynomial function: \(f(x) = 4x^3 + 6x^2 - 2x - 1\), and the known zero: \(x = \frac{1}{2}\).
2. Use polynomial division or synthetic division to divide \(f(x)\) by the factor corresponding to the known zero, which is \(\left(x - \frac{1}{2}\right)\).
3. Perform the division to find the quotient polynomial, which will be a quadratic polynomial since the original is cubic.
4. Set the quotient polynomial equal to zero and solve for the remaining zeros using the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a\), \(b\), and \(c\) are coefficients from the quadratic.
5. List all zeros: the known zero \(\frac{1}{2}\) and the two zeros found from solving the quadratic equation.

Key Concepts

Polynomial Zeros and Roots

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

Polynomial Division (Synthetic or Long Division)

Polynomial division is used to divide a polynomial by a binomial corresponding to a known zero, simplifying the polynomial to a lower degree. Synthetic division is a shortcut method when dividing by linear factors, making it easier to find remaining zeros.

Factoring and Solving Quadratic Equations

After dividing the polynomial, the quotient is often a quadratic that can be factored or solved using the quadratic formula. This step is essential to find the remaining zeros once the polynomial is reduced to degree two.