Question

1.) Find all zeros of the function f(x)=6x3−7x2−14x+8. and 2.) Given y=3x5−x4+9x3−3x2−12x+4, and that 2i is a zero, write y in factored form (as a product of linear factors). Be sure to write the full equation

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73 Answers

1. To find the zeros of f(x)=6x^3−7x^2−14x+8 , apply the Rational Root Theorem and synthetic division:

- Test potential rational roots:\pm 1, \pm 2, \pm 4, \pm 8, \pm \frac{1}{2}, \pm \frac{1}{3}, \pm \frac{2}{3}, \pm \frac{4}{3}, \pm \frac{8}{3} .

- Using synthetic division, we find thatf\left(\frac{-2}{3}\right) = 0 and f(2) = 0 .

- Further factorization shows:

(x + \frac{2}{3})(x - 2)(6x + 4) = 0

- Simplify to find the zeros:

x = -\frac{2}{3}, 2

Answer:x = -\frac{2}{3}, 2

2. To writey=3x^5−x^4+9x^3−3x^2−12x+4 in factored form given that 2i is a zero:

- Since2i is a zero, -2i is also a zero due to complex conjugates appearing together.

- Using synthetic division, check additional real roots.

- Suppose1, 2, -1 are real zeros (use synthetic division to verify).

- Factor together:

3(x - 2i)(x + 2i)(x - 1)(x - 2)(x + 1)

Answer:y = 3(x - 2i)(x + 2i)(x - 1)(x - 2)(x + 1)

- Test potential rational roots:

- Using synthetic division, we find that

- Further factorization shows:

- Simplify to find the zeros:

Answer:

2. To write

- Since

- Using synthetic division, check additional real roots.

- Suppose

- Factor together:

Answer:

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