cubic polynomials

Question

How do I use synthetic division to factor cubic polynomials like this

x^3 - 5x^2 - 2x + 24

Answer 1

Given polynomial x^3-5x^2-2x+24

By synthetic division

x^3-5x^2-2x+24 = (x+2)(x^2-7x+12)

x^2-4x-3x+12

x(x-4)-3(x-4)

(x-4)(x-3)

Factoring of given polynomial is (x+2)(x-4)(x-3)

Answer 2

The polynomial function x³- 5x²- 2x + 24

From factor theorem when f(c) = 0 then x - c is factor of polynomial.

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- 5x²- 2x + 24 = 0

If p/q is a rational zero, then p is a factor of 24 and q is a factor of 1.

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

The possible values for q are ± 1.

So, p/q =   ± 1, ±2,  ± 3, ±4, ±6, ±8, ±12 and   ±24.

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

p/q	1	-5	-2	24
1	1	-4	-6	30
-1	1	-6	4	20
-2	1	-7	12	0

Since f(-2)  =  0,  x   =  -2 is a zero. The depressed polynomial is   x² - 7x + 12 = 0

Since the depressed polynomial of this zero,  x² - 7x + 12, is quadratic, use the  Factorization find the roots of the related quadratic equation

 x² - 7x + 12 = 0

 x²- 4x - 3x + 12 = 0

x(x - 4) - 3(x - 4) = 0

(x - 4) (x - 3) = 0

x - 4 = 0 and x - 3 = 0

x = 4 and x = 3

Zeros of the cubic polynomial are x = 3, 4 and -2.

Factoring of x³- 5x²- 2x + 24 is (x - 4)(x - 3)(x + 2).