Formula used:

The factors of a polynomial can be find by taking a common factor and this method is called factor by grouping,

\(ab+ac+bd+cd=a(b+c)+d(b+c)\)

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

Or,

\(ab-ac+bd-cd=a(b-c)+d(b-c)\)

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

Calculation:

Consider the polynomial \(x^{12} + x^{7} +x^{5} +1\).

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

\(x^{12}+x^{7}+x^{5}+1 = (x^{12}+x^{7})(x^{5} +1)\)

\(=x^{7}(x^{5}+1)+1(x^{5}+1)\)

As, \((x^{5} + 1)\) is the common factor of the polynomial,

The polynomial can be factorized as,

\(x^{7}(x^{5}+1)+1(x^{5}+1)=(x^{5}+1)(x^{7}+1)\)

Therefore, the factorization of the polynomial \(x^{12} + x^{7} + x^{5} + 1\) is \((x^{5} +1) (x^{7} +1)\).

The factors of a polynomial can be find by taking a common factor and this method is called factor by grouping,

\(ab+ac+bd+cd=a(b+c)+d(b+c)\)

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

Or,

\(ab-ac+bd-cd=a(b-c)+d(b-c)\)

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

Calculation:

Consider the polynomial \(x^{12} + x^{7} +x^{5} +1\).

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

\(x^{12}+x^{7}+x^{5}+1 = (x^{12}+x^{7})(x^{5} +1)\)

\(=x^{7}(x^{5}+1)+1(x^{5}+1)\)

As, \((x^{5} + 1)\) is the common factor of the polynomial,

The polynomial can be factorized as,

\(x^{7}(x^{5}+1)+1(x^{5}+1)=(x^{5}+1)(x^{7}+1)\)

Therefore, the factorization of the polynomial \(x^{12} + x^{7} + x^{5} + 1\) is \((x^{5} +1) (x^{7} +1)\).