find the roots of the polynomial equation x^3-2x^2+10x+136=0

Question

I need help figuring out how to find the roots of this polynomial equation.

Answer 1

Given polynomial equation x^3-2x^2+10x+136 = 0

By synthatic division

 

Answer 2

Identify Rational Zeros :

Usually it is not practical to test all possible zeros of a polynomial function using only synthetic substitution. The Rational Zero Theorem can be used for finding the some possible zeros to test.

Rational Root Theorem, if a rational number in simplest form p/q is a root of the polynomial equation a_nx ⁿ + a_n – 1x ^{n – 1} + ... + a₁x + a₀ = 0, then p is a factor of a₀ and q is a factor if a_n.

The function is x ³- 2x ²+ 10x + 136 = 0.

If p/q is a rational zero, then p is a factor of 136 and q is a factor of 1.

The possible values of p are   ± 1,   ± 4, and ± 34.

The possible values for q are ± 1.

By the Rational Roots Theorem, the only possible rational roots are, p/q = ± 1,   ± 4, and ± 34.

Make a table for the synthetic division and test possible real zeros.

Make a table for the synthetic division and test possible real zeros.

p/q	1	- 2	10	136
1	1	- 1	9	145
2	1	0	10	156
3	1	1	13	175
-1	1	- 3	13	123
- 2	1	- 4	18	100
- 3	1	- 5	25	61
- 4	1	- 6	34	0

Since, f(- 4) = 0, x = - 4 is a zero. The depressed polynomial is  x ² - 6x + 34 = 0

Answer 3

Since the depressed polynomial of this zero, x²-2x+5, is quadratic, use the Quadratic Formula to find the roots of the related quadratic equation.

x = [ - b ± √(b ² - 4ac ) ] / 2a.

Substitute a = 1, b = - 6, and c = 34.

x = [ - (- 6) ± √((- 6)² - 4 * 1 * 34) ] / 2 * 1

x = [ 6 ± √(36 - 136) ] / 2

x = [ 6 ± √(- 100) ] / 2

x = [ 6 ± 10i ] / 2

⇒ x = 3 ± 5i.

Therefore, the roots of the function are x = - 4, x = 3 + 5i., and x = 3 - 5i.