If x3+3px+q has a factor of the form x2-2ax+a2, then q2+4p3=0.

I want to see the answer pl

I want to see the answer

Answer please

Answer

Answer pls

To prove that q^2 + 4p^3 = 0 when x^3 + 3px + q has a factor of the form x^2 - 2ax + a^2, we can use polynomial division and factorization.

Let's assume that x^2 - 2ax + a^2 is a factor of x^3 + 3px + q. When we perform polynomial division, we divide x^3 + 3px + q by x^2 - 2ax + a^2, and the remainder should be zero if it is a factor. So we have:

x^3 + 3px + q = (x^2 - 2ax + a^2)(x + bx + c) + 0

Expanding the right side and comparing coefficients, we get:

x^3 + 3px + q = (x^3 + (b - 2a)x^2 + (2ab - a^2 + c)x + (-2ac + a^2 + bc))

Comparing coefficients of corresponding powers of x, we have:

1. x^3 coefficient: 1 = 1 ⟹ a = 1 (equation 1)

2. x^2 coefficient: 0 = b - 2a ⟹ 0 = b - 2 ⟹ b = 2 (equation 2)

3. x^1 coefficient: 3p = 2ab - a^2 + c (equation 3)

4. x^0 (constant) coefficient: q = -2ac + a^2 + bc (equation 4)

Using equations 1 and 2, we substitute the values of a and b in equations 3 and 4:

3p = 2(1)(2) - (1)^2 + c ⟹ 3p = 4 - 1 + c ⟹ c = 3p - 3 (equation 5)

q = -2(1)(3p - 3) + (1)^2 + 2(3p - 3) ⟹ q = -6p + 6 + 1 + 6p - 6 ⟹ q = 1

Now that we have found q = 1, let's substitute this value in the equation q^2 + 4p^3 = 0:

(1)^2 + 4p^3 = 1 + 4p^3 = 0

Therefore, we have proved that q^2 + 4p^3 = 0 when x^3 + 3px + q has a factor of the form x^2 - 2ax + a^2.