f(x)=x^4 -x^3 -6x^2 +4x +8 rational Zeros Theorem

Question

using the rational zeros theorem to find all real zeros of the polynomial function.

Answer 1

The zeros of this function are  x = -1, -2 and 2f (x) = x⁴ - x³- 6x² + 4x + 8

If p/q is a rational zero, then p is a factor of 8 and q is a factor of 1.

The possible values of p are   ± 1,   ± 2, and   ± 4.

The possible values for q are ± 1.

So, p/q = ± 1,   ± 2, ± 4.

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

p/q	1	- 1	- 6	4	8
1	1	0	- 6	- 2	6
2	1	1	- 4	- 4	0

Since f(2) = 0, you know that x = 2 is a zero. The depressed polynomial is x³  +  x² - 4x + 4

Factor x³  +  x² - 4x + 4

x³  +  x² - 4x + 4 = 0

The possible values of p are   ± 1,   ± 2

The possible values for q are ± 1.

So, p/q = ± 1, ± 2

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

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

Since f(2) = 0, you know that x = 2 is a zero. The depressed polynomial is  x² + 3x + 2

Factor x² + 3x + 2

x² + 3x + 2 = 0

x² + 2x + x + 2 = 0

x(x +2) + (x + 2) = 0

(x + 2) (x + 1) = 0

Apply Zero Product Property

(x + 2) = 0 or (x + 1) = 0

x = -2 , x = -1

The zeros of this function are  x = -1, -2 and 2

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.

f (x) = x⁴ - x³- 6x² + 4x + 8

If p/q is a rational zero, then p is a factor of 8 and q is a factor of 1.

The possible values of p are   ± 1,   ± 2, and   ± 4.

The possible values for q are ± 1.

So, p/q = ± 1,   ± 2, ± 4.

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

p/q	1	- 1	- 6	4	8
1	1	0	- 6	- 2	6
2	1	1	- 4	- 4	0
-1	1	- 2	-4	0	0
-2	1	-3	0	0	0

Since f (2) = 0, f (-1) = 0 and f (- 2) = 0 then x = - 1,x = - 2 and x = 2 are zeros.

f (- 2) = 0, the depressed polinomial is x⁴ - 3x³= 0

x³(x - 3) = 0

( x - 3) = 0 → x = 3 ia a zero.

The function has real zeros at x = -1, x = - 2, x = 2 and x = 3.