Question

Find a polynomial​ p(x) with real coefficients having a degree​ 6, leading coefficient 33​, and zeros 44​, 0​ (multiplicity 3), and 44minus−33i.

Answer 1

Given zero "a", (x - a) is a factor of the polynomial. That factor is repeated according to the multiplicity of root "a". Any complex zero is matched by another zero that is its conjugate. Hence all 6 zeros are defined, and our polynomial is

... p(x) = 3(x -4)(x -0)³((x -4)² -(3i)²)

... p(x) = 3x³(x -4)(x² -8x +25) = 3x³(x³ -12x² +57x -100)

... p(x) = 3x⁶ -36x⁵ +171x⁴ -300x³

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The factor pair (x - a - bi)(x - a + bi) can be treated as the factorization of the difference of squares (x - a)² - (bi)². When expanded, that becomes

... x² -2ax + a²+b²