To find the quadratic polynomial with zeros summing up to 5/2 and the product of the zeros being 1, we can use the fact that the sum of the zeros of a quadratic polynomial is equal to the negative coefficient of the linear term divided by the coefficient of the quadratic term, while the product of the zeros is equal to the constant term divided by the coefficient of the quadratic term.

Let's denote the quadratic polynomial as ax^2 + bx + c, where a, b, and c are constants.



The sum of the zeros is given as 5/2, so we have:
- b/a = 5/2

Multiplying both sides by 'a', we get:
b = (5/2)a



The product of the zeros is given as 1, so we have:
c/a = 1

Multiplying both sides by 'a', we get:
c = a



Now, we can substitute the values of b and c in terms of 'a' into the quadratic polynomial:
ax^2 + ((5/2)a)x + a

Simplifying, we get:
2ax^2 + 5ax + 2a

So, the quadratic polynomial is 2ax^2 + 5ax + 2a.



To find the zeros of the polynomial 2ax^2 + 5ax + 2a, we can use the splitting of the middle term method.

First, let's factor out 'a' from the polynomial:
a(2x^2 + 5x + 2)

To split the middle term, we need to find two numbers whose product is 2 * 2 = 4 and whose sum is 5.

The numbers that satisfy these conditions are 4 and 1.

So, we can rewrite the middle term as:
a(2x^2 + 4x + x + 2)

Now, we can group the terms and factor by grouping:
a[(2x^2 + 4x) + (x + 2)]
a[2x(x + 2) + 1(x + 2)]
a[(2x + 1)(x + 2)]

Therefore, the zeros of the polynomial 2ax^2 + 5ax + 2a are the values of x that make (2x + 1) and (x + 2) equal to zero.

For (2x + 1) = 0, x = -1/2.
For (x + 2) = 0, x = -2.

So, the zeros of the polynomial are x = -1/2 and x = -2.