A quadratic polynomial is a polynomial of degree 2 in the form of f(x) = ax² + bx + c, where a, b, and c are constants. The zeros or roots of a polynomial are the values of x for which f(x) equals zero. In other words, when we substitute the zeros into the polynomial, the resulting expression evaluates to zero.



To find the zeros of a quadratic polynomial, we can set the polynomial equal to zero and solve for x. For the polynomial f(x) = ax² + bx + c, the zeros can be found by solving the equation ax² + bx + c = 0.



Given that alpha and beta are zeros of the quadratic polynomial f(x) = ax² + bx + c, we need to evaluate alpha²beta and alpha*beta².

Let's denote the zeros as alpha and beta, respectively.

α = alpha
β = beta

We know that the sum of the zeros of a quadratic polynomial is given by the formula alpha + beta = -b/a, and the product of the zeros is given by the formula alpha * beta = c/a.

Now, let's evaluate alpha²beta:

α²β = (α * α) * β
= α² * β

Similarly, let's evaluate alpha*beta²:

αβ² = α * (β * β)
= α * β²

Now, substituting the values from the formulas for the sum and product of the zeros, we get:

α²β = (α * α) * β
= (-b/a * -b/a) * (c/a)
= (b²/a²) * (c/a)
= b²c/a³

Similarly,

αβ² = α * (β * β)
= (-b/a) * (c/a * c/a)
= (-b/a) * (c²/a²)
= -bc²/a³

Therefore, alpha²beta = b²c/a³ and alpha*beta² = -bc²/a³.



- Given a quadratic polynomial f(x) = ax² + bx + c with zeros alpha and beta.
- We need to evaluate alpha²beta and alpha*beta².
- Using the formulas for the sum and product of the zeros, we substitute the values and simplify to find the results.
- alpha²beta = b²c/a³
- alpha*beta² = -bc²/a³

What is the value of f(x)