Factoring help?

Question

Factor the expression 

X^3 - X^2 - 30x + 72

Answer 1

The polynomial x ³ - x ² - 30x + 72

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- x²- 30x + 72

If p/q is a rational zero, then p is a factor of 72 and q is a factor of 1.

The possible values of p are   ± 1,   ± 2,  ± 3, ± 4, ± 6, ± 8, ± 9,± 12, ±24, ± 36.  .

The possible values for q are ± 1.

So, p/q = ± 1, ± 2,  ± 3.

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

p/q	1	-1	-30	72
1	1	0	-30	42
2	1	1	-28	16
3	1	2	-24	0

Since f (3)  =  0,   x  =  3 is a zero. The depressed polynomial is   x ² + 2x - 24 = 0

From the factor theorem

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

So (x - 3) is a factor of the polynomial.

Now factorise x ² + 2x - 24 = 0

x ² + 6x - 4x - 24 = 0

x (x  + 6) - 4(x  + 6) = 0

(x  + 6) (x  - 4) = 0

Apply zero product property.

x + 6 = 0 and x - 4 = 0

x  = -6 and x  = 4

Zeros are x  = 3, 4, -6 then the factors are (x - 3), (x - 4) and (x + 6).

Factoring of x ³ - x ² - 30x  + 72  = (x  - 3) (x  - 4) (x + 6).