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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 $$(x -1) (x - 2)$$.


Solution

Using Division Algorithm here:-
$$Dividend=Divisor\times Quotient+Remainder$$

So, Applying it$$:-$$
Let $$q(x),k(x)$$ be quotient when $$f(x)$$ is divided by $$x-1$$ and $$x-2 $$ respectively

$$\Rightarrow f(x)=(x-1)q(x)+5   $$
$$\therefore f(1)=5$$ ..... $$(1)$$

Also,$$f(x)=(x-2)k(x)+7$$
$$\therefore f(2)=7$$ ..... $$(2)$$

Now, let $$ax+b$$ be remainder when $$f(x)$$ is divided by $$(x-1)(x-2)$$ and $$g(x)$$ be quotient.
$$f(x)=(x-1)(x-2)g(x)+(ax+b)$$
Using $$(1)$$ and $$(2)$$
$$5=a+b$$ ...... $$(3)$$
$$7=2a+b$$ ...... $$(4)$$

Solving $$(3)$$ and $$(4)$$, we get
$$a=2$$ and $$b=3$$
$$\therefore 2x+3$$ is remainder when $$f(x)$$ is divided by $$(x-1)(x-2).$$

Mathematics

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