1. Q has degree 3 and zeros 4, 4i, and −4i
2. R has degree 4 and zeros 3 − 2i and 1, with 1 a zero of multiplicity 2.
3. U has degree 5, zeros 1/2 ,−3, and −i, and leading coefficient 4; the zero −3 has multiplicity 2.
2. R has degree 4 and zeros 3 − 2i and 1, with 1 a zero of multiplicity 2.
3. U has degree 5, zeros 1/2 ,−3, and −i, and leading coefficient 4; the zero −3 has multiplicity 2.
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1. Q has factors (x - 4), (x - 4i), and (x + 4i). Expanded, the polynomial is
(x-4)(x-4i)(x+4i)
(x-4)(x² +16)
x³ - 4x² + 16x - 64
2. Factors: (3 - 2i), which implies (3 + 2i), and (x-1), which has mult 2, so it's (x-1)².
The first two factors appear from applying the quadratic equation to (x² - 6x + 13).
(x² - 6x + 13)(x - 1)²
(x² - 6x + 13)(x² - 2x + 1)
x^4 - 6x³ + 13x² - 2x³ + 12x² - 26x + x² - 6x + 13
x^4 - 8x³ + 25x² - 32x + 13
3. Factors: (x - 1/2), (x + 3)². The zero at -i implies a zero at i, which can only be from the polynomial: (x² - 2x + 5/4)
4 (x² - 2x + 5/4) (x + 3)² (x - 1/2)
4 (x² - 2x + 5/4) (x² + 6x + 9)(x - 1/2)
(x-4)(x-4i)(x+4i)
(x-4)(x² +16)
x³ - 4x² + 16x - 64
2. Factors: (3 - 2i), which implies (3 + 2i), and (x-1), which has mult 2, so it's (x-1)².
The first two factors appear from applying the quadratic equation to (x² - 6x + 13).
(x² - 6x + 13)(x - 1)²
(x² - 6x + 13)(x² - 2x + 1)
x^4 - 6x³ + 13x² - 2x³ + 12x² - 26x + x² - 6x + 13
x^4 - 8x³ + 25x² - 32x + 13
3. Factors: (x - 1/2), (x + 3)². The zero at -i implies a zero at i, which can only be from the polynomial: (x² - 2x + 5/4)
4 (x² - 2x + 5/4) (x + 3)² (x - 1/2)
4 (x² - 2x + 5/4) (x² + 6x + 9)(x - 1/2)