Question

Obtain the zeroes of the polynomial p(x) = 2x² – 5x – 3. Hence, obtain a polynomial each of whose zeroes is one less than each of the zero of p(x).

Answer 1

Given: p(x) = 2x² – 5x – 3.

Step-wise calculation:

1. Factor p(x):

2x² – 5x – 3 = (2x + 1)(x – 3)

2. Set p(x) = 0 to get zeros:

2x + 1 = 0

⇒ `x = -1/2` 

And x – 3 = 0

⇒ x = 3 

So the zeros are 3 and `-1/2`.

3. New zeros are each one less than these:

3 − 1 = 2 and `(-1/2) − 1 = -3/2`

4. Form a quadratic with roots 2 and `-3/2`:

`(x - 2)(x + 3/2) = x^2 - (1/2)x - 3` 

Clearing denominators gives 2x² – x – 6 (multiply by 2).

Zeros of p(x): x = 3 and x = `-1/2`. A polynomial whose zeros are one less than those of p(x) is `x^2 - (1/2)x - 3` or with integer coefficients, 2x² – x – 6.