use rational zero theorem , find all zeros

Question

Use the Rational Zeros theorem to find all real zeros of each polynomial function. Use the zeros to factor f over the real numbers.

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

 

Answer 1

Step 1:

The function f (x ) = x ³ + 2x ² - 5x  - 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.

f (x ) = x ³ + 2x ² - 5x  - 6

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 and  ± 6.

The possible values for q are ± 1.

So, p /q = ± 1,  ± 2, ± 3 and  ± 6.

Step 2:

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

p /q	1	2	-5	-6
1	1	3	-2	-8
-1	1	1	-6	0

Since f (-1) = 0,   x = -1 is a zero. The depressed polynomial is   x² + x - 6 = 0

Since the depressed polynomial of this zero, x² + x - 6, is quadratic, use the Factorization to find the roots of the related quadratic equation.

 x² + x - 6 = 0

 x² + 3x - 2x - 6 = 0

x ( x  + 3) - 2(x  + 3) = 0

(x  + 3) (x  - 2) = 0

Apply zero product property.

x  + 3 = 0 and x  - 2 = 0

x  = -3 and x  = 2

Zeros of f (x ) = x ³ + 2x ² - 5x  - 6 are x  = -1 , -3 and 2.

By using Factor theorem,

When f (c ) = 0 then x  - c  is a factor of polynomial.

Factoring of f (x ) = (x  - 2)(x  + 3)(x  + 1).

Solution :

Zeros are x  = -1 , -3 and 2.