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?
+121 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
\(r(x)=x\)
