Solve the equation 2x3−3x2−11x+6=0 given that -2 is a zero of f(x)=2x³−3x²−11x+6.

Verified step by step guidance

1. Since -2 is a zero of the polynomial $f\left(x\right)=2x^3-3x^2-11x+6$, use synthetic division or polynomial division to divide $f(x)$ by $(x\; +\; 2)$ to find the quotient polynomial.
2. Set up synthetic division with the coefficients of $f(x)$: 2, -3, -11, and 6, and use -2 as the divisor.
3. Perform the synthetic division step-by-step: bring down the first coefficient, multiply by -2, add to the next coefficient, and repeat until all coefficients are processed. This will give you the coefficients of the quotient polynomial, which will be a quadratic.
4. Write the quotient polynomial from the synthetic division result in the form $ax^2\; +\; bx\; +\; c$.
5. Solve the quadratic equation obtained from the 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

A zero or root of a polynomial is a value of x that makes the polynomial equal to zero. Knowing a zero helps factor the polynomial and find other roots. For example, if -2 is a zero of f(x), then (x + 2) is a factor of f(x).

Polynomial Division

Polynomial division, such as synthetic or long division, is used to divide a polynomial by a factor corresponding to a known zero. This process simplifies the polynomial to a lower degree, making it easier to solve for remaining zeros.

Factoring and Solving Cubic Equations

After dividing out a known factor, the resulting polynomial can often be factored further or solved using quadratic methods. Factoring breaks the polynomial into simpler expressions set to zero, allowing the identification of all roots of the cubic equation.