Pre Calc questions 3

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Mathematics

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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.

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Explanation & Answer

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1. 𝑓(π‘₯ ) = π‘₯ 3 βˆ’ 5π‘₯ 2 + 6π‘₯ βˆ’ 30.
First, guess an integer root among divisors of the free term, 30.
It is clear that π‘₯ 1 = 5 is a root: 𝑓(5) = 53 βˆ’ 53 + 30 βˆ’ 30 = 0.
Second, divide 𝑓 (π‘₯ ) by π‘₯ βˆ’ π‘₯ 1 :
π‘₯ 3 βˆ’ 5π‘₯ 2 + 6π‘₯ βˆ’ 30 = π‘₯ 2 (π‘₯ βˆ’ 5) + 6(π‘₯ βˆ’ 5) = (π‘₯ 2 + 6)(π‘₯ βˆ’ 5).
Now we see that 𝑓(π‘₯ ) has no more (real) roots because π‘₯ 2 + 6 β‰₯ 6 > 0 for any π‘₯.
The answer: the only real root of 𝑓(π‘₯ ) is π‘₯ = πŸ“.

2. To do this, substitute οΏ½...


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