Question

When a polynomial $f(x)$ is divided by $(x - 1),$ the remainder is $5$ and when it is divided by $(x - 2),$ the remainder is $7.$ find the remainder when it is divided by $(x - 1)(x - 2).$

Answer 1

Hint: Convert the given word statements in the form of mathematics and then find the function. Once the function is found then divide it with the given dividend and simplify for the resultant value.

Complete step by step answer:
Using the Division Algorithm which states that –
Dividend $ = $Divisor $ \times $Quotient $ + $Remainder
Now, applying the same concept –
Let q(x), k(x) be the quotient when the function f(x) is divided by $(x - 1)$ and $(x - 2)$ respectively.
$ \Rightarrow f(x) = (x - 1)q(x) + 5$
Place $x = 1$in the above equation
$ \Rightarrow f(1) = (1 - 1)q(x) + 5$
Simplify the above equation-
$ \Rightarrow f(1) = 5$ …… (A)
$ \Rightarrow f(x) = (x - 2)q(x) + 7$
Similarly, Place $x = 2$in the above equation
$ \Rightarrow f(2) = (2 - 2)q(x) + 7$
Simplify the above equation-
$ \Rightarrow f(2) = 7$ …… (B)
Now, let the remainder be $ax + b$ when the $f(x)$ is divided by $(x - 1)(x - 2)$ and quotient be $g(x)$
$f(x) = (x - 1)(x - 2)g(x) + (ax + b)$
Using the equations (A) and (B)
$
  5 = a + b\;{\text{ }}.....{\text{ (C)}} \\
  7 = 2a + b\;{\text{ }}.....\;{\text{(D)}} \\
 $
Subtract equation (C) from (D)
Here the left side of both the equations are subtracted and the right hand side of both the sides of the equations are subtracted.
$7 - 5 = (2a + b) - (a + b)$
When there is a negative sign outside the bracket then the sign of all the terms inside the bracket changes when the bracket is opened.
$ \Rightarrow 2 = 2a + b - a - b$
Make the pair of like terms in the above expression
$ \Rightarrow 2 = \underline {2a - a} + \underline {b - b} $
Simplify the above expression. Like terms with the same value and opposite sign cancel each other.
$ \Rightarrow 2 = a$
Place the above value in equation (C)
$
   \Rightarrow 5 = a + b \\
   \Rightarrow 5 = 2 + b \\
   \Rightarrow b = 3 \\
 $
Hence,
$a = 2$ and $b = 3$

Hence, $(2x + 3)$ is the remainder when the function, $f(x)$ is divided by $(x - 1)(x - 2)$

Note:Be careful while simplification of the equations and the sign convention. Always remember when there is a negative sign outside the bracket the sign of the terms inside the bracket changes. Positive terms become negative and negative becomes positive.