The roots of the entire polynomic expression, that is, the product of p(x) = x^2 + 8x + 12 and q(x) = x^3 + 5x^2 - 6x, are x₁ = 0, x₂ = -2, x₃ = -3 and x₄ = -6.
A value of x is said to be a root of the polynomial if and only if r(x) = 0. Let be r(x) = p(x) · q(x), then we need to find the roots both for p(x) and q(x) by factoring each polynomial, the factoring is based on algebraic properties:
r(x) = (x + 6) · (x + 2) · x · (x² + 5 · x - 6)
r(x) = (x + 6) · (x + 2) · x · (x + 3) · (x + 2)
r(x) = x · (x + 2)² · (x + 3) · (x + 6)
By direct inspection, we conclude that the roots of the entire polynomic expression are x₁ = 0, x₂ = -2, x₃ = -3 and x₄ = -6. [tex]\blacksquare[/tex]
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