## Description

1.Find all the zeroes of of the polynomial function f(x)=x^3-5x^2+6x-30. If you use synthetic division show all three lines of numbers. Be specific!

2. Use the Remainder Theorem to determine if x-2 is a factor of the polynomial f(x)=3x^5-7x^3-11x^2+2.

3.Describe the behavior of the graph at the x-intercepts for the function f(x)=(2x-7)^7(x+3)^4. Be sure to identify each x-intercept and justify your answer.

## Explanation & Answer

The solutions are ready, please ask if something is unclear.docx and pdf files are identical

1. 𝑓(𝑥 ) = 𝑥 3 − 5𝑥 2 + 6𝑥 − 30.

First, guess an integer root among divisors of the free term, 30.

It is clear that 𝑥 1 = 5 is a root: 𝑓(5) = 53 − 53 + 30 − 30 = 0.

Second, divide 𝑓 (𝑥 ) by 𝑥 − 𝑥 1 :

𝑥 3 − 5𝑥 2 + 6𝑥 − 30 = 𝑥 2 (𝑥 − 5) + 6(𝑥 − 5) = (𝑥 2 + 6)(𝑥 − 5).

Now we see that 𝑓(𝑥 ) has no more (real) roots because 𝑥 2 + 6 ≥ 6 > 0 for any 𝑥.

The answer: the only real root of 𝑓(𝑥 ) is 𝑥 = 𝟓.

2. To do this, substitute �...

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