Inyalan0

2021-12-10

Factor each polynomial completely. If the polynomial cannot be factored, say it is prime.

${x}^{4}+{x}^{3}+x+1$

Medicim6

Beginner2021-12-11Added 33 answers

Step 1

Given:

Let$p={x}^{4}+{x}^{3}+x+1$

To Find: To factorise the given polynomial

Step 2

Solution:

$p={x}^{4}+{x}^{3}+x+1$

$p={x}^{3}(x+1)+1(x+1)$

$p=({x}^{3}+1)(x+1)$

$p=({x}^{3}+{1}^{3})(x+1)$

We know that${a}^{3}+{b}^{3}=(a+b)({a}^{2}-ab+{b}^{2})$

Hence$p=(x+1)[{x}^{2}-\left(x\right)\left(1\right)+{1}^{2}](x+1)$

$p=(x+1)({x}^{2}-x+1)(x+1)$

$p=(x+1)(x+1)({x}^{2}-x+1)$

Hence${x}^{4}+{x}^{3}+x+1=(x+1)(x+1)({x}^{2}-x+1)$

Given:

Let

To Find: To factorise the given polynomial

Step 2

Solution:

We know that

Hence

Hence