Question

Find the zeros of the following quadratic polynomial and verify the relationship between the zeros and the coefficients:

`p(x) = x^2 + 4/3x - 4/3`

Answer 1

Given: `p(x) = x^2 + (4/3)x - (4/3)`.

Step-wise calculation:

1. Clear fractions:

3p(x) = 3x² + 4x – 4

2. Factor 3x² + 4x – 4:

3x² + 4x – 4 = (3x – 2)(x + 2)

3. Solve (3x – 2)(x + 2) = 0:

3x – 2 = 0

⇒ `x = 2/3` 

x + 2 = 0

⇒ x = –2

So the zeros are `α = 2/3` and β = –2.

4. Verify relationship with coefficients for p(x) = ax² + bx + c `("here"  a = 1, b = 4/3, c = -4/3)`:

Sum: `α + β = 2/3 + (-2)`

= `2/3 - 6/3`

= `-4/3`

= `-b/a`

Product: `αβ = (2/3)(-2)`

= `-4/3`

= `c/a`

Zeros are `x = 2/3` and x = –2. They satisfy `α + β = -b/a = -4/3` and `αβ = c/a = -4/3`, verifying the standard relations between zeros and coefficients.