Question

How many roots of the polynomial 30x^4 + 7x^3-125x^2-54x+72 have absolute value greater than 1?

Answer 1

Answer:

There are 3 roots of the given polynomial that have absolute value greater than 1.  

Step-by-step explanation:

We are given the polynomial:

$p(x) = 30x^4 + 7x^3-125x^2-54x+72$

We can factorize the given polynomial as:

$p(x) = 30x^4 + 7x^3-125x^2-54x+72\\=(x-2)(2x+3)(5x-3)(3x+4)=0$

$\text{Solving for}\\(x-2) = 0\\x = 2\\|x| = 2 > 1$

$\text{Solving for}\\(2x+3) = 0\\x = \dfrac{-3}{2}\\\bigg|\dfrac{-3}{2}\bigg| = \dfrac{3}{2} > 1$

$\text{Solving for}\\(5x-3) = 0\\x = \dfrac{3}{5}\\\bigg|\dfrac{3}{5}\bigg| = \dfrac{3}{5} < 1$

$\text{Solving for}\\(3x+4) = 0\\x = \dfrac{-4}{3}\\\bigg|\dfrac{-4}{3}\bigg| = \dfrac{4}{3} > 1$

Thus, there are 3 roots of the given polynomial that have absolute value greater than 1.