Find the zeros of the following quadratic polynomials and verify the basic relationships between the zeros and the coefficients.

3x² – 5x + 2

Let f(x) = 3x² – 5x + 2.

To find out zeros of the given polynomial.

We put f(x) = 0

⇒ 3x² – 5x + 2 = 0

To find out roots of this polynomial we use splitting of middle term method.

According to this method we need to find two numbers whose sum is – 5 and product is 6.

∴ 3x² – (3 + 2)x + 2 = 0

⇒3x² – 3x – 2x + 2 = 0

⇒ 3x(x – 1) – 2(x – 1) = 0

⇒ (3x – 2)(x – 1) = 0

When, 3x – 2 = 0

Then, x = .

Again when, x – 1 = 0

∴ then, x = 1

⇒ Our zeros are α = and β = 1.

⇒ sum of zeros = α + β = + 1

⇒ sum of zeros = α + β =

⇒ Product of zeros = αβ = .

Now, Comparing f(x) = 3x² – 5x + 2 with standard equation ax² + bx + c.

We get, a = 3, b = – 5 and c = 2.

We can verify,

⇒ Sum of zeros =

i.e. α + β =

∴ α + β =

⇒ Product of zeros =

αβ =

Hence, relationship between zeros and coefficient is verified.