Need to calculate:The factorization of 6z3+3z2+2z+1.

Question

SchulzD · Accepted Answer

Formula used:
The factors of a polynomial can be found by taking a common factor and this method is called factor by grouping,

a b + a c + b d + c d = a (b + c) + d (b + c)

= (a + d) (b + c)

Or,

a b - a c + b d - c d = a (b - c) + d (b - c)

= (a + d) (b - c)

Calculation:
Consider the polynomial

6 z^{3} + 3 z^{2} + 2 z + 1

.
This is a four term polynomial, factorization of this polynomial can be found by factor by grouping as,

6 z^{3} + 3 z^{2} + 2 z + 1 = (6 z^{3} + 3 z^{2}) + (2 z + 1)

= 3 z^{2} (2 z + 1) + 1 (2 z + 1)

As,

(2 z + 1)

is the common factor of the polynomial,
The polynomial can be factorized as,

6 z^{3} + 3 z^{2} + 2 z + 1 = 3 z^{2} (2 z + 1) + 1 (2 z + 1)

= 2 z + 1) (3 z^{2} + 1)

Therefore, the factorization of the polynomial is

(2 z + 1) (3 z^{2} + 1)

.

Need to calculate:The factorization of 6z^{3}+3z^{2}+2z+1.