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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