Question

Find the zeroes of the polynomial p(x) = 4x² – 8x + 3 and verify the relationship between its zeroes and co-efficients.

Answer 1

1. Find the zeroes

To find the zeroes, set the polynomial p(x) = 0 and solve for x using the factorization method (splitting the middle term):

1. Identify coefficients: For 4x² – 8x + 3, the coefficients are a = 4, b = –8 and c = 3.

2. Split the middle term: We need two numbers that multiply to a × c = 12 and add to b = – 8. These numbers are –6 and –2.

3. Factorize:

4x² – 6x – 2x + 3 = 0

2x(2x – 3) – 1(2x – 3) = 0

(2x – 1)(2x – 3) = 0

4. Solve for x:

1. 2x – 1 ⇒ `x = 1/2`

2. 2x – 3 ⇒ `x = 3/2`

2. Verify the relationship

Let `α = 1/2` and `β = 3/2`.

Sum of zeroes

The formula for the sum of zeroes is α + β

= `- ("coefficent of"  x)/("coefficient of"  x^2)`

= `- b/a`

LHS: `α + β = 1/2 + 3/2`

= `4/2`

= 2

RHS: `- b/a`

= `- ((-8))/4`

= `8/4`

= 2

LHS = RHS, hence verified.

Product of zeroes

The formula for the product of zeroes is α × β

= `("constant term")/("coefficient of"  x^2)`

= `c/a`

LHS: `α xx β = 1/2 xx 3/2`

= `3/4`

RHS: `c/a = 3/4`

LHS = RHS, hence verified.