find all roots of 4x414x336x284x72Solution4x414x336x284x720

find all roots of 4x^4-14x^3+36x^2-84x+72

Solution

4x^4-14x^3+36x^2-84x+72=0

Rational Root Theorem tells you that if the polynomial has a rational zero then it must be a fraction p/q, where p is a factor of the trailing constant and q is a factor of the leading coefficient.

The factors of the leading coefficient (4) are 1, 2 ,2 ,4 .The factors of the constant term (72) are 1, 2 ,3 ,4, 6 ,8 ,9 ,12, 18, 24 ,36, 72 . Then the Rational Roots Tests yields the following possible solutions:

±1/1, ±1/2, ±1/2, ±1/4, ±2/1, ±2/2, ±2/2, ±2/4, ±3/1, ±3/2, ±3/2, ±3/4, ±4/1, ±4//2, ±42, ±4/4, ±6/1, ±6/2, ±6/2, ±6/4

±8/1, ±8/2, ±8/2, ±8/4, ±9/1, ±9/2, ±9/2, ±9/4, ±12/1, ±12/2, ±12/2, ±12/4, ±18/1, ±18/2, ±18/2,

±18/4, ±24/1, ±24/2, ±24/2, ±24/4, ±36/1, ±36/2, ±36/2, ±36/4, ±72/1, ±72/2, ±72/2, ±72/4

we plug all these values into the polynomial P(x), we obtain P(2)=0.

So,To find remaining zeros we use Factor Theorem. This theorem states that if pq is root of the polynomial then this polynomial can be divided with qxp.

(4x^4-14x^3+36x^2-84x+72)/(x-2) = 4x^36x^2+24x36.

Now we find the remaining roots of this cubic polynomial: 4x^36x^2+24x36.

Rational Root Test- The factors of the leading coefficient (4) are 1, 2 ,2 ,4 .The factors of the constant term (-36) are 1, 2, 3, 4, 6, 6, 9, 12, 18, 36 . Then the Rational Roots Tests yields the following possible solutions:

If we plug these values into the polynomial P(x), we obtain P(3/2)=0

So, x= 3/2 is one of the roots of polynomial

Again by Factor theorem : Divide polynomial by 2x -3

(4x^36x^2+24x36)/( 2x-3) = 2x^2 +12

Now solve 2x^2 +12=0 ---> x= +/- isqrt6 ( imaginary roots)

So, the roots of the polynomail 4x^4-14x^3+36x^2-84x+72 are : x= 2 , 3/2 , + isqrt6 , -isqrt6