Question

findall rational zeros of the polynomial p(x)=x^4-x^3-5x^2+3x+6.

Answer 1

p(x) = x^4-x^3-5x^2+3x+6

By synthatic division

x^4-x^3-5x^2+3x+6 = (x+1)(x-2)(x^2-3)

Now x^2-3 = 0

x^2 = 3

x = √3

x = ±1.732

All rational zeros of given polynomial is x = -1,2,±1.732

Answer 2

P (x) = x⁴ - x³ - 5x² + 3x + 6

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

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, ± 3.

The possible values for q are ± 1

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

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

p/q	1	- 1	- 5	3	6
1	1	0	- 5	- 2	4
2	1	1	- 3	- 3	0

Since f(2)  =  0,  x  =  2 is a zero. The depressed polynomial is  x³ + x² – 3x - 3 = 0.

If p/q is a rational zero, then p is a factor of  3 and q is a factor of 1.

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

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

p/q	1	1	- 3	- 3
1	1	2	- 1	- 4
3	1	4	9	24
-1	1	0	-3	0

Since f(-1) = 0, x = - 1 is a zero.The depressed polynomial is  x² – 3= 0

Since the depressed polynomial of this zero, x² – 3 = 0.

Factor x² – 3

x² – 3 = 0

x² – ()² = 0

Apply formula : (a ² - b ²) = (a + b)(a - b).

(x  + )(x  -) = 0

Apply Zero Product Property.

(x  + ) = 0 (or) (x  -) = 0

x =  - , x = .

The rational zeros of the given polynomial are  at x = - 1, 2, , and  - .