When the polynomial f(x) is divided by (x-3) the remainder is 15. When f(x) is divided by (x-1)^2 the remainder is 2x+1 Find the remainder when f(x) is divided by (x-3)(x-1)^2
According to the Remainder Theorem, when a polynomial p(x) is divided by x−a, the remainder is p(a). So, when f(x) is divided by (x−3), the remainder is f(3), and we are given that f(3)=15.
Similarly, when f(x) is divided by (x−1)2, the remainder is f(1), and we are given that f(1)=2(1)+1=3.
We can write the remainder when f(x) is divided by (x−3)(x−1)2 as R(x). By the Remainder Theorem, we have the following:
R(3)=f(3)=15
R(1)=f(1)=3
To find R(x), we can use the polynomial long division algorithm to divide f(x) by (x−3)(x−1)2. The long division algorithm gives us the following:
Quotient: x+5 Divisor: (x-3)(x-1)^2 ---------------------- x^3-4x^2+7x-15 -(x^3-3x^2) -x^2+7x-15 +x^2-x^2 +7x-15-(-7x+21) -x-15+21 -6 ---------------------- x+5
Therefore, the remainder when f(x) is divided by (x−3)(x−1)2 is x+5.
So the answer is x+5.
