Solve left(x^2+51right)left(x^2-91right)left(x+51right)^3

$$\left( { x }^{ 2 } +51 \right) \left( { x }^{ 2 } -91 \right) { \left(x+51 \right) }^{ 3 }$$

Evaluate

$$\left(x^{2}-91\right)\left(x^{2}+51\right)\left(x+51\right)^{3}$$

Solution Steps

Use binomial theorem \left(a+b\right)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3} to expand \left(x+51\right)^{3}.

$$\left(x^{2}+51\right)\left(x^{2}-91\right)\left(x^{3}+153x^{2}+7803x+132651\right)$$

Use the distributive property to multiply x^{2}+51 by x^{2}-91 and combine like terms.

$$\left(x^{4}-40x^{2}-4641\right)\left(x^{3}+153x^{2}+7803x+132651\right)$$

Use the distributive property to multiply x^{4}-40x^{2}-4641 by x^{3}+153x^{2}+7803x+132651 and combine like terms.

$$x^{7}+153x^{6}+7763x^{5}+126531x^{4}-316761x^{3}-6016113x^{2}-36213723x-615633291$$

Expand

$$x^{7}+153x^{6}+7763x^{5}+126531x^{4}-316761x^{3}-6016113x^{2}-36213723x-615633291$$

Solution Steps

Use binomial theorem \left(a+b\right)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3} to expand \left(x+51\right)^{3}.

$$\left(x^{2}+51\right)\left(x^{2}-91\right)\left(x^{3}+153x^{2}+7803x+132651\right)$$

Use the distributive property to multiply x^{2}+51 by x^{2}-91 and combine like terms.

$$\left(x^{4}-40x^{2}-4641\right)\left(x^{3}+153x^{2}+7803x+132651\right)$$

Use the distributive property to multiply x^{4}-40x^{2}-4641 by x^{3}+153x^{2}+7803x+132651 and combine like terms.

$$x^{7}+153x^{6}+7763x^{5}+126531x^{4}-316761x^{3}-6016113x^{2}-36213723x-615633291$$