help factoring polynomials

Question

Can you help me factor this polynomial?
x^3 + 4x^2 + 5x + 2
I don't understand how to factor larger polynomials unless they are factorable by grouping. Can someone give me some general steps to help me understand? Thanks.

Answer 1

Given polynomial is x^3+4x^2+5x+2

For x = -1

x^3+4x^2+5x+2 = (-1)^3+4(-1)^2+5(-1)+2

= -1+4-5+2

= -6+6

= 0

So (x+1) is a factor of given polynomial.

By synthetic division

x^3+4x^2+5x+2 = (x+1)(x^2+3x+2)

Now factorize x^2+3x+2

= x^2+2x+x+2

= x(x+2)+1(x+2)

= (x+2)(x+1)

Factoring of x^3+4x^2+5x+2 is (x+1)(x+2)(x+1).

Answer 2

The polynomial x³+ 4x²+ 5x + 2

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+ 4x²+ 5x + 2 = 0

If p/q is a rational zero, then p is a factor of 2 and q is a factor of 1.

The possible values of p are   ± 1, and  ± 2.

The possible values for q are ± 1.

So, p/q = ± 1,   ± 2.

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

p/q	1	4	5	2
1	1	2	1	16
-1	1	3	2	0

Since f(-1)   =   0,   x   =   -1 is a zero. The depressed polynomial is   x²+ 3x + 2 = 0

Since the depressed polynomial of this zero,  x²- 3x + 2 , is quadratic, use the Factorisation to find the factors of the related quadratic equation.

x² + 3x + 2 = 0

x²+ 2x + x + 2 = 0

x ( x + 2) + 1 (x + 2) = 0

(x + 2)(x + 1) = 0

Apply zero product property.

x = -2 and x = -1

Zeros of the polynomial are x = -1, -2 and -1.

Factoring of x³+ 4x²+ 5x + 2 = (x + 1)(x + 1)(x + 2).