Solve the factorization of x^{12}+x^{7}+x^{5}+1

Question
Polynomial factorization
asked 2020-11-02
Solve the factorization of \(x^{12}+x^{7}+x^{5}+1\)

Answers (1)

2020-11-03
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 \((x^{5} +1) (x^{7} +1)\).
0

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