Find the LCM of polynomials $p (x) = x^{3} - 1$ , $q (x) = x^{3} + 1$ and $r (x) = {(x^{2} + 1)}^{2} - x^{2}$ .

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Solution

We observe $p (x) = x^{3} - 1 = x^{3} - 1^{3}$ . Factoring using $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ , we get $p (x) = (x - 1) (x^{2} + x + 1)$
Similarly, $q (x) = x^{3} + 1 = x^{3} + 1^{3} = (x + 1) (x^{2} - x + 1)$ ; we have used $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ .
Finally $r (x) = {(x^{2} + 1)}^{2} - x^{2} = (x^{2} + x + 1) (x^{2} - x + 1)$ .
irreducible factors of $p (x)$ or $q (x)$ Highest exponent
Algebraic $(x + 1)$ $1$
Algebraic $(x - 1)$ $1$
Algebraic $(x^{2} - x + 1)$ $1$
Algebraic $(x^{2} + x + 1)$ $1$
Hence the LCM
$(x + 1) (x - 1) (x^{2} + x + 1) (x^{2} - x - 1)$

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(4)t(y) = $y^{4} + y + 5$

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irreducible factors of $p (x)$ or $q (x)$	Highest exponent
Algebraic $(x + 1)$	$1$
Algebraic $(x - 1)$	$1$
Algebraic $(x^{2} - x + 1)$	$1$
Algebraic $(x^{2} + x + 1)$	$1$