Question

Find the value of a, if \[x + 1\] is a factor of \[{x^3} + a\].

Answer 1

Hint: If \[x + 1\] is a factor then the polynomial will become zero if we take \[x + 1\] as an equation and substitute the value of x we can obtain from \[x + 1 = 0\].

Complete Step by Step Solution:
Let us try to find out the value of x from \[x + 1 = 0\]
\[\begin{array}{l}
\therefore x + 1 = 0\\
 \Rightarrow x = - 1
\end{array}\]
Now as we have the value of x let us put it in \[{x^3} + a = 0\] to get the value of a.
As we know that the value obtained from the factor of a polynomial will make the polynomial zero. In other words it is a root of the polynomial so while putting that value we can write it as 0.
\[\begin{array}{l}
\therefore {x^3} + a = 0\\
 \Rightarrow {\left( { - 1} \right)^3} + a = 0\\
 \Rightarrow - 1 + a = 0\\
 \Rightarrow a = 1
\end{array}\]
So from all of these we are getting the value of a as 1.

Note: We can also check whether it's true or not by putting the value of a. After putting the whole thing in, it becomes \[{x^3} + 1\] . By applying \[{a^3} + {b^3} = (a + b)({a^2} - ab + {b^2})\]
\[\therefore {x^3} + 1 = (x + 1)({x^2} - x + 1) = 0\]
So clearly \[x + 1\] is a root of \[{x^3} + 1\]
So the answer satisfies.