Question

According to the rational root theorem, which of the following are possible

roots of the polynomial function below? Check all that apply.
x4-6x³-19x2 +84x+180
A. 19
B. 8
C. 10
D. 18

Answer 1

C, D

Explanation:

You want to know possible rational roots of x⁴-6x³-19x² +84x+180.

Rational root theorem

The rational root theorem tells you any rational roots will be fractions of the form ...

±(divisor of the constant)/(divisor of the leading coefficient)

Application

Here, the coefficient of x⁴, the leading coefficient, is 1. So any rational roots will be divisors of ±180. Those divisors are (plus or minus) ...

1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, 180

The highlighted values are ones that are answer choices.

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Additional comment

An estimate of the upper bound of the magnitude of the roots would be ...

`2\max(\sqrt[4]{180},\sqrt[3]{84},√(19),6)=12`

The rational root theorem allows roots larger than this.

The roots of this polynomial are actually {-3, -2, 5, 6}.