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

4x² + 8x

Let f(x) = 4x² + 8x

Arranging equation in proper form.

Now, f(x) = 4x² + 8x + 0

To find out zeros of the given polynomial.

We put f(x) = 0

⇒ f(x) = 4x² + 8x + 0 = 0

∴4x² + 8x = 0

⇒4x(x + 2) = 0

Now, 4x = 0

∴ x = 0

When, (x + 2) = 0

Then, x = – 2

⇒ Our zeros are α = 0 and β = – 2.

⇒ sum of zeros = α + β = 0 + ( – 2)

⇒ sum of zeros = α + β = – 2.

⇒ Product of zeros = αβ = 0 × ( – 2) = 0.

Now, Comparing f(x) = 4x² + 8x + 0 with standard equation ax² + bx + c.

We get, a = 4, b = 8 and c = 0.

We can verify,

⇒ Sum of zeros =

i.e. α + β =

∴ α + β = – 2

⇒ Product of zeros =

αβ =

Hence, relationship between zeros and coefficient is verified.