Question

Using the remainder theorem, find the remainder when $7{x^3} + 40{x^2} + 22x - 35$ is divided by $(x + 1)$ ?

Answer 1

Hint: The remainder theorem is an approach of Euclidean division of polynomials. The remainder theorem states that if a polynomial $P(x)$ is divided by a factor $(x - a)$, then the remainder is $P(a)$. Using this concept we will find $P(a)$ by substituting x=a.

Complete step by step answer:
The remainder theorem states that if a polynomial $P(x)$ is divided by a factor $(x - a)$, then the remainder is $P(a)$.
The dividend polynomial is $7{x^3} + 40{x^2} + 22x - 35$ and the divisor factor is $(x + 1)$.
The linear factor is $(x + 1)$.
Comparing it with $(x - a)$, we get $(x - ( - 1))$.
Therefore, $a = - 1$.
Now, we will get the reminder by finding $P( - 1)$.
$P(x)$ = $7{x^3} + 40{x^2} + 22x - 35$
According to remainder theorem,
Reminder = $P( - 1)$
Substituting $x = - 1$ in the given polynomial,
Reminder = $7{( - 1)^3} + 40{( - 1)^2} + 22( - 1) - 35$
Simplifying the cube and square part,
Reminder = $7( - 1) + 40(1) + 22( - 1) - 35$
Eliminating the brackets,
Reminder = $ - 7 + 40 - 22 - 35$
Simplifying the terms,
Reminder = $ - 24$
Therefore, we get the reminder as $ - 24$ when $7{x^3} + 40{x^2} + 22x - 35$ is divided by $(x + 1)$.

Note:
In this above problem, when dividing $7{x^3} + 40{x^2} + 22x - 35$ by $(x + 1)$, we can say that $(x + 1)$ is one of the factors of the dividend polynomial. Remainder theorem is an alternative method to long division method and it’s only useful for finding reminders. The advantage of the remainder theorem is it is short and very convenient to find the reminder whereas the disadvantage is it can only be used to find a reminder and not quotient.