Q. 1 A4.6( 5 Votes )

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

x^{2} – 2x – 8

Answer :

Let f(x) = x^{2} – 2x – 8

To find out zeros of the given polynomial.

We put f(x) = 0

⇒ x^{2} – 2x – 8 = 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 8.

∴ x^{2} – (4 – 2)x – 8 = 0

⇒ x^{2} – 4x + 2x – 8 = 0

⇒ x(x – 4) + 2(x – 4) = 0

⇒ (x + 2)(x – 4) = 0

∴ x = – 2 and x = 4.

⇒ Our zeros are α = – 2 and β = 4.

⇒ sum of zeros = α + β = – 2 + 4 = 2.

⇒ Product of zeros = αβ = ( – 2) × 4 = – 8.

⇒ Comparing f(x) = x^{2} – 2x – 8 with standard equation ax^{2} + bx + c = 0.

We get, a = 1, b = – 2 and c = – 8

We can verify,

⇒ Sum of zeros =

i.e. α + β =

∴ α + β = 2

⇒ Product of zeros =

αβ =

αβ = – 8.

Hence, relationship between zeros and coefficient is verified.

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