SOLUTION: Let P(x)=x4 −2x3 −10x2 +6x+45 ▪ Use the Rational Zero Theorem to list all the possible rational zeros. ▪ Then find all zeros exactly (rational, irrational, and imaginary

Question 1156241: Let P(x)=x4 −2x3 −10x2 +6x+45
▪ Use the Rational Zero Theorem to list all the possible rational zeros.
▪ Then find all zeros exactly (rational, irrational, and imaginary).
Hint: Use the Rational Zero Theorem, a graphing calculator, and synthetic division if needed.
Answer by ikleyn(52780) (Show Source): You can put this solution on YOUR website!
.


Since the leading coefficient is 1, the Remainder theorem provides this list of possible zeros 
(all of them are divisors of the constant term 45, in this case)

   +/-1, +/-3, +/-5, +/-9, +/-15, +/-45.



Next, the plot below


    


    Plot y =

shows the root x= 3 of the multiplicity at least 2.

So, I divide by , and I get the quotient .

This quotient is a quadratic polynomial with negative discriminant, so it has no real roots.

Therefore, factoring over real numbers is

= .

You may go forward to find complex zeroes of the quadratic quotient.

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