how do you solve p(x)=3x^4-8x^3-6x^2+17x+6

Question

a. list all the possible rational zeros

b. find the rational zeros.

Answer 1

The polinomial is P (x ) = 3x⁴ - 8x³ - 6x² + 17x + 6.

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_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 3.

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

The possible values for q are ± 1, ± 3.

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

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

p/q	3	- 8	- 6	17	6
1	3	- 2	- 10	- 3	0
2	3	- 2	- 10	- 3	0

Since f(2) = 0, x = 2 is a zero. The depressed polynomial is  3x³ - 2x² – 10x - 3 = 0.

Answer 2

Contd.....

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

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

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

p/q	3	- 2	- 10	- 3
1	3	1	- 9	- 12
- 1	3	- 6	- 4	1
-1/3	3	- 3	- 9	0

Since f(- 1/3) = 0, x = - 1/3 is a zero. The depressed polynomial is  3x² – 3x - 9 = 0.

Devide each side by 3.

x² – x - 3 = 0.

Since the depressed polynomial of this zero, x² – x - 3, is quadratic, use the Quadratic Formula to find the roots of the related quadratic equationx² – x - 3.

x = [-b ± √(b^2 - 4ac)]/2a

Substitute b = - 1, a = 1, and c = - 3.

x = [-(- 1) ± √((- 1)^2 - 4 * 1 * (- 3))]/2 * 1

x = [ 1 ± √(1 + 12)]/2

x = [ 1 ± √(13)]/2

x = [ 1 ± 3.6]/2

x = [ 1 + 3.6]/2 and x = [ 1 - 3.6]/2

x = 4.6/2 and x = - 2.6/2

x = 2.3 and x = - 1.3 .

The rational zeros of the given polinomail are x = 2, 2.3, - 1.3, and - 1/3.