find roots of g(x)= x^4-3x^3+5x^2-27x-36

Question

find all the roots.

Answer 1

g(x) = x^4-3x^3+5x^2-27x-36

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 y = x ⁴ - 3x ³ + 5x ²- 27x - 36.

If p/q is a rational zero, then p is a factor of - 36 and q is a factor of 1.

The possible values of p are   ± 1, ± 4, ± 9.

The possible values for q are ± 1.

By the Rational Roots Theorem, the only possible rational roots are, p/q = ± 1, ± 4, ± 9.

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

p/q	1	- 3	5	- 27	- 36
1	- 2	3	- 2	- 24	- 60
2	1	- 1	3	- 21	- 78
- 1	1	- 4	9	- 36	0

Since, f(- 1) = 0, x = - 1 is a zero. The depressed polynomial is  x ³ - 4x ² + 9x - 36 = 0.

Answer 3

If p/q is a rational zero, then p is a factor of - 36 and q is a factor of 1.

By the Rational Roots Theorem, the only possible rational roots are, p/q = ± 1, ± 4, ± 9.

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

p/q	1	- 4	9	- 36
1	1	- 3	6	- 30
2	1	- 2	5	- 26
3	1	- 1	6	- 18
4	1	0	9	0

Since, f(4) = 0, x = 4 is a zero. The depressed polynomial is  x ² + 9 = 0.

x ² = - 9

x ² = (± 3i )²

Extract square roots from each side.

⇒ x = ± 3i.

Therefore, tthe roots of the function are x = - 1, x = 4, x = 3i, and x = - 3i.