Question

If the sum of the zeroes of the polynomial p(x) = (p + 1)x² + (2p + 3)x + (3p + 4) is –1, then find the value of p.

Answer 1

Given:

Polynomial p(x) = (p + 1)x² + (2p + 3)x + (3p + 4).

Sum of its zeroes is –1.

Step-wise calculation:

1. For a quadratic ax² + bx + c, sum of zeroes = `-b/a`.

2. Here a = p + 1 and b = 2p + 3, so sum = `-(2p + 3)/(p + 1)`.

3. Set this equal to –1: `-(2p + 3)/(p + 1) = -1` 

⇒ `(2p + 3)/(p + 1) = 1`

⇒ 2p + 3 = p + 1

⇒ p = –2

4. Check validity: a = p + 1

= –2 + 1

= –1 ≠ 0

So the polynomial is quadratic for p = –2.