Question

Find the factor of$ - {a^8} + {b^8}$.

Answer 1

Hint: Factors are the numbers which are multiplied with the other factors gives another number. For example, $2 \times 3 = 6$, here, $2$ and $3$ are the factors, which when multiplied with each other gives another number. To find the factors of $ - {a^8} + {b^8}$, firstly, we need to simplify it with the help of identity.

Formula used: $ {x^2} - {y^2} = (x + y)(x - y)$

Complete step-by-step solution:
To find the factors of $ - {a^8} + {b^8}$, we will simplify it by using identity. The identity is
$ {x^2} - {y^2} = (x + y)(x - y).....................................(a)$
We can also write the given equation as,
$
   \Rightarrow b{}^8 - {a^8} \\
   \Rightarrow {({b^4})^2} - {({a^4})^2}...........................................................(1) \\
 $
Now, this equation looks exactly like the identity. Now, we will substitute the value into the identity (a),
$ \Rightarrow {({b^4})^2} - {({a^4})^2} = ({b^4} + {a^4})({b^4} - {a^4}).....................................................(2)$
We can write $({b^4} - {a^4})$ as ${({b^2})^2} - {({a^2})^2}$. Now, after substituting this into identity (a), we get,
$ \Rightarrow {({b^2})^2} - {({a^2})^2} = ({b^2} + {a^2})({b^2} - {a^2})..............................................(3)$
Now, substitute this equation $(3)$ in equation $(2)$,
$ \Rightarrow {({b^4})^2} - {({a^4})^2} = ({b^4} + {a^4})({b^2} + {a^2})({b^2} - {a^2}).....................................................(4)$
We can write $ \Rightarrow {b^2} - {a^2} = (b + a)(b - a)...................................................................(5)$
Now, substitute equation$(5)$in equation $(4)$, we get,
$ \Rightarrow {({b^4})^2} - {({a^4})^2} = ({b^4} + {a^4})({b^2} + {a^2})(b + a)(b - a)..........................................................................(6)$
And we know that, $b{}^8 - {a^8} = {({b^4})^2} - {({a^4})^2}$, so we write the equation$(6)$ as,
$ \Rightarrow {b^8} - {a^8} = ({b^4} + {a^4})({b^2} + {a^2})(b + a)(b - a)$
Hence, the factors of $ - {a^8} + {b^8}$ are $({b^4} + {a^4}),({b^2} + {a^2}),(b + a)$ and $(b - a)$, which when multiplied with each other gives the required number.

Note: We can also check that the factors we have found are correct or not, by multiplying them with each other results into the given equation. We can check it by using identity or by simply multiplying it but the process of multiplying is much longer, so it is easy to check with the help of the identity. We can also check if the identity we have given is true or not, by multiplying the factors of the identity and then we will find that the right hand side is equal to the left hand side.