Question

Medal!!! find a third degree polynomial equation with rational coefficients that has roots -4 and 6+i

Answer 1

A theorem similar to the Rational Roots Theorem states that if a polynomial with real coefficients has a root of an imaginary number, then another one of its roots must be the conjugate.

Given these two roots, -5 and 6 + i, we also have 6 - i as a root.

So, since these are all roots of the same function, we simply multiply them all together as so:
(x + 5)(x - (6 + i))(x - (6 - i))
With the associative property of addition, we can rewrite this into
= (x + 5)((x - 6) + i)((x - 6) - i)
= (x + 5)((x - 6)² - (i)²
Remember that i = √-1, so i² = -1
= (x + 5)(x² - 12x + 36 - 1)
= (x + 5)(x² - 12x + 35)
= x³ - 12x² + 35x + 5x² - 60x + 175
= x³ - 7x² - 25x + 175

The equation is y = x³ - 7x² - 25x + 175
I hope this helps!

Answer 2

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