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When a certain polynomial is divided by x - 2, the remainder is 2.  When the polynomial is divided by x + 2, the remainder is -2.  What is the remainder when the polynomial is divided by x^2 - 4?

Dec 19, 2019

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Let's say the certain polynomial is $$f(x)$$. Then

$$f(x)=g_1(x)\cdot(x-2)+2$$, and

$$f(x)=g_2(x)\cdot(x+2)-2$$.

so

$$f(2)=g_1(2)\cdot(2-2)+2=2$$, and

$$f(-2)=g_2(-2)\cdot(-2+2)-2=-2$$

If we divide $$f(x)$$ by $$x^2-4$$, we end up with a quotient polynomial $$g(x)$$ and remainder $$r(x)$$,

where $$r(x)$$ is a linear function of x, since it's degree is1 less that the degree of the divisor polynomial. So

we can write

$$f(x)=g(x)\cdot(x^2-4)+r(x)$$, where $$r(x)=mx+b$$.

Now evaluate the function $$f$$ at 2 and -2:

$$f(2)=g(2)\cdot(2^2-4)+r(2)=0+r(2)=2$$, which implies

$$r(2)=2m+b=2$$, and

$$f(-2)=g(-2)\cdot((-2)^2-4)+r(-2)=r(-2)$$, which means

$$r(-2)=-2m+b=-2$$.

By eliminating b from the above two equations we get $$m=1$$ and $$b=0$$; that gives us the remainder

$$r(x)=x$$

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Dec 19, 2019