Question

Identify all rational zeros of the polynomial function f(x)= x^3+4x^2-2x-15.

Answer 1

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) = x³ + 4x² - 2x - 15 = 0

If p/q is a rational zero, then p is a factor of - 15 and q is a factor of 1.

The possible values of p are   ± 1,   ± 3, and   ± 5.

The possible values for q are ± 1.

So, p/q = ± 1,   ± 3, ± 5.

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

p/q	1	4	-2	-15
1	1	5	-	-12
2	1	6	10	5
-3	1	1	-5	0

Since f(-3)   =   0,   x   =   -3 is a zero. The depressed polynomial is   x² + x - 5 = 0

Since the depressed polynomial of this zero, x² + x - 5 = 0, is quadratic, use the Quadratic Formula to find the roots of the related quadratic equation

The function has one real zero at x = -3 and two real zeros at .