Question

Find the following product: $({x^2} - 1)({x^4} + {x^2} + 1)$

Answer 1

Hint: According to the question we have to determine the product or multiplication of $({x^2} - 1)({x^4} + {x^2} + 1)$. So, to find the product first of all we have to use the formula as given below:
$ \Rightarrow ({a^3} - {b^3}) = {a^3} - {a^2}b + {a^2}b - a{b^2} + a{b^2} - {b^3}....................(A)$
Now, we have to arrange the terms of the formula by multiplying each term with the other terms to obtain in the form of $(a - b)({a^2} + {b^2} + ab)$
Now, we have to compare the obtained formula with the given expression to find the product.

Complete step-by-step answer:
Step 1: First of all we have to use the formula (A) as mentioned in the solution hint. Hence,
$ \Rightarrow ({a^3} - {b^3}) = {a^3} - {a^2}b + {a^2}b - a{b^2} + a{b^2} - {b^3}$
Now, on solving and rearranging the terms of the expression as obtained just above,
$ \Rightarrow ({a^3} - {b^3}) = ({a^3} - {a^2}b) + ({a^2}b - a{b^2}) + (a{b^2} - {b^3})$…………….(1)
Step 2: Now, we have to take out the common terms from the expression (1) as obtained in the solution step 1. Hence,
$ \Rightarrow ({a^3} - {b^3}) = {a^2}(a - b) + ab(a - b) + {b^2}(a - b)..................(2)$
Step 3: Now, as we can see that in the expression (2) as obtained in the solution step 2 we can take common $(a - b)$to simplify it. Hence,
$ \Rightarrow ({a^3} - {b^3}) = (a - b)({a^2} + {b^2} + ab)..................(3)$
Step 4: Now, we have to compare the expression (3) as obtained in the solution step with the expression given in the question, Hence,
$(a - b)$is $({x^2} - 1)$and $({a^2} + {b^2} + ab).$is $({x^4} + {x^2} + 1)$
Step 5: Now, with the help of expression (3) we can write the expression $({x^2} - 1)({x^4} + {x^2} + 1)$in form of $(a - b)({a^2} + {b^2} + ab)$as obtained in the solution step 4. Hence,
$ = ({x^2} - 1)({(x)^2} + {x^2}(1) + {1^2})$
But as we know that from the solution step 3
$ \Rightarrow ({a^3} - {b^3}) = (a - b)({a^2} + {b^2} + ab)$Hence,
$
   = ({({x^2})^3} - {1^3}) \\
   = {x^6} - 1
 $

Hence, with the help of formula (A) above we have obtained the product of $({x^2} - 1)({x^4} + {x^2} + 1)$$ = {x^6} - 1$

Note: Alternate method: We can find the product of given expression $({x^2} - 1)({x^4} + {x^2} + 1)$ by some simply multiplication of each terms as given below:
Step 1: First of all we have multiply the terms $({x^2} - 1)$ with the terms $({x^4} + {x^2} + 1)$hence,
\[
   = ({x^2} - 1) \times ({x^4} + {x^2} + 1) \\
   = {x^2} \times ({x^4} + {x^2} + 1) - 1 \times ({x^4} + {x^2} + 1) \\
   = {x^6} + {x^4} + {x^2} - {x^4} - {x^2} - 1..............(B)
 \]
Step 2: Now, we have to eliminate the terms of the expression (B) as obtained in the solution step 1. Hence,
$ = {x^6} - 1$
So, we have obtained the product of $({x^2} - 1)({x^4} + {x^2} + 1)$$ = {x^6} - 1$