x^3 + 2x^2 - 5x - 6.

Question

se the Rational Zero Theorem to find all the real zeros. Use the zeros to factor f over the real numbers?

Answer 1

The function f(x) = x³ + 2x² - 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.

x³ + 2x² - 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, ± 3 and  ± 6.

The possible values for q are ± 1.

So, p/q = ± 1,  ± 2, ± 3 and  ± 6.

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

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

x + 3 = 0 and x - 2 = 0

x = - 3 and x = 2

Real zeros are - 1 , - 3 and 2.

From Factor theorem,

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

Factoring of x³ + 2x² - 5x - 6 = (x + 1)(x + 3)(x - 2).