Solve the equation 12x³+16x²−5x−3=0 given that -3/2 is a root.

Verified step by step guidance

1. Since \(-\frac{3}{2}\) is a root of the polynomial \(12x^{3} + 16x^{2} - 5x - 3 = 0\), use polynomial division or synthetic division to divide the cubic polynomial by the factor \(\left(x + \frac{3}{2}\right)\).
2. To perform synthetic division, rewrite the root \(-\frac{3}{2}\) as \(-1.5\) and set up the coefficients of the polynomial: 12, 16, -5, and -3.
3. Carry out the synthetic division process step-by-step to find the quotient polynomial, which will be a quadratic expression.
4. Once you have the quadratic quotient, set it equal to zero and solve for \(x\) using the quadratic formula: \[x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\], where \(a\), \(b\), and \(c\) are the coefficients of the quadratic.
5. Combine the root \(-\frac{3}{2}\) with the solutions from the quadratic to write the complete solution set for the original cubic equation.

Key Concepts

Polynomial Roots and Factor Theorem

The Factor Theorem states that if a polynomial f(x) has a root r, then (x - r) is a factor of f(x). Given that -3/2 is a root, (x + 3/2) must be a factor of the polynomial, which helps in factoring and simplifying the equation.

Polynomial Division

Polynomial division, either long division or synthetic division, is used to divide the original polynomial by the factor corresponding to the known root. This process reduces the polynomial's degree, making it easier to solve the remaining equation.

Solving Quadratic Equations

After factoring out the known root, the remaining polynomial is quadratic. Solving this quadratic equation using methods like factoring, completing the square, or the quadratic formula yields the other roots of the original cubic equation.