Questions: Solve the equation 16x^3+16x^2-x-1=0 given that -1/4 is a zero of f(x)=16x^3+16x^2-x-1

Solution Steps

To solve the equation \(16x^3 + 16x^2 - x - 1 = 0\) given that \(-\frac{1}{4}\) is a zero of the polynomial, we can use polynomial division to factor out \((x + \frac{1}{4})\) from the polynomial. This will reduce the polynomial to a quadratic equation, which we can then solve using the quadratic formula.

Given that \(-\frac{1}{4}\) is a zero of the polynomial \(16x^3 + 16x^2 - x - 1\), we can factor out \((x + \frac{1}{4})\) from the polynomial. This reduces the polynomial to a quadratic equation.

After performing polynomial division, the quotient is a quadratic equation. Solving this quadratic equation yields the other roots of the polynomial.

The solutions to the polynomial equation \(16x^3 + 16x^2 - x - 1 = 0\) are the roots of the quadratic equation along with the given zero \(-\frac{1}{4}\).

Final Answer