Find the Zeroes of the Polynomial x^2 + x – p(p+1)

Find the Zeroes of the Polynomial x^2 + x – p(p+1)

Solution:

Let’s go through a more detailed solution to find the zeros of the polynomial x^2 + x – p(p+1).

To find the zeroes, we set the polynomial equal to zero:

x^2 + x – p(p+1) = 0

Now, we’ll use the quadratic formula to solve for x:

x = (-b ± √(b^2 – 4ac)) / (2a)

Comparing our equation to the standard form ax^2 + bx + c = 0,

we have:

a = 1
b = 1
c = -p(p+1)

Substituting these values into the quadratic formula, we get:

x = (-(1) ± √[(1)^2 – 4(1)(-p(p+1)))] / (2(1))

x = (-1 ± √(1 + 4p(p+1))) / 2

x = (-1 ± √(1 + 4p^2 + 4p)) / 2

x = (-1 ± √(4p^2 + 4p + 1)) / 2

Simplifying further, we have:

x = (-1 ± √((2p + 1)^2)) / 2

x = (-1 ± (2p + 1)) / 2

Now, we can split this equation into two separate equations:

x = (-1 + (2p + 1)) / 2x
= (2p) / 2
x = p

x = (-1 – (2p + 1)) / 2
x = (-2 – 2p) / 2
x = -1 – p

Therefore, the zeros of the polynomial x^2 + x – p(p+1) are x = p and x = -1 – p.

Answer:

Zeros of the Polynomial x^2 + x – p(p+1)

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