Question

Find the zeroes of the polynomial p(x) = 3x² – 4x – 4. Hence, write a polynomial whose each of the zero is 2 more than zeroes of p(x).

Answer 1

Given: p(x) = 3x² – 4x – 4.

Step-wise calculation:

1. Find zeroes of p(x) using the quadratic formula:

`x = (4 ± sqrt((-4)^2 - 4 xx 3 xx (-4)))/(2 xx 3)` 

= `(4 ± sqrt(16 + 48))/6` 

= `(4 ± sqrt(64))/6` 

= `(4 ± 8)/6` 

So `x_1 = (4 + 8)/6` 

= `12/6`

= 2 

And `x_2 = (4 - 8)/6` 

= `(-4)/6`

= `(-2)/3`

2. Each new zero is 2 more than the original zeroes:

New zeroes = 2 + 2 = 4 and `(-2/3) + 2 = 4/3`

3. Form the quadratic with roots 4 and `4/3`: 

`(x - 4)(x - 4/3) = x^2 - (16/3)x + 16/3` 

Multiply by 3 to clear denominators: 3x² – 16x + 16.

Zeroes of p(x) are 2 and `-2/3`. A polynomial whose each zero is 2 more than those of p(x) is 3x² – 16x + 16.