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

2x² – 2√2 x + 1

Let f(x) = 2x² – 2x + 1

To find out zeros of the given polynomial.

We put f(x) = 0

⇒ 2x² – 2x + 1 = 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 – 2 and product is 2.

∴ 2x² – ()x + 1 = 0

⇒2x² – x – x + 1 = 0

⇒ x(x – 1) – 1(x – 1) = 0

⇒ (x – 1)(x – 1) = 0

⇒ (x – 1)² = 0

∴ x = ,

⇒ Our zeros are α = and β = .

⇒ sum of zeros = α + β = +

⇒ sum of zeros = α + β =

⇒ Product of zeros = αβ = .

Now, Comparing f(x) = 2x² – 2x + 1 with standard equation ax² + bx + c.

We get, a = 2, b = – 2 and c = 1.

We can verify,

⇒ Sum of zeros =

i.e. α + β =

∴ α + β =

⇒ Product of zeros =

αβ =

Hence, relationship between zeros and coefficient is verified.