Question

Find all roots of x^3-2x^2-5x+6.

Answer 1

Given polynomial x^3-2x^2-5x+6

By synthetic division

x^3-2x^2-5x+6 = (x-1)(x+2)(x-3)

Roots are x = 1,-2,3.

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- 5x + 6 = 0.

If p/q is a rational zero, then p is a factor of 6 and q is a factor of 1.

The possible values of p are   ± 1, ± 2, and ± 3.

The possible values for q are ± 1.

By the Rational Roots Theorem, the only possible rational roots are, p / q = ± 1,   ± 2, and ± 3.

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	- 5	6
- 1	1	- 3	8	- 2
2	1	0	- 5	- 4
- 3	1	- 5	10	- 9
1	1	- 1	- 6	0

Since, f(1) = 0, x = 1 is a zero. The depressed polynomial is  x ² - x - 6 = 0.

By factor by grouping.

x ² - 3x + 2x - 6 = 0

x(x - 3) + 2(x - 3) = 0

Factor : (x - 3)(x + 2) = 0

Apply zero product property.

x - 3 = 0 and x + 2 = 0

⇒ x = 3 and x = - 2.

Therefore, tthe roots of the function are x = 1, x = - 2, and x = 3.