To find the remainder when the polynomial 5x3 + 7x + 5 is divided by x2, we can use the polynomial remainder theorem.
According to this theorem, when dividing a polynomial by x2, the remainder will be a polynomial of degree less than 2. Thus, the remainder can be expressed in the form Ax + B, where A and B are constants.
We can find the remainder by substituting values for x. It’s easiest to choose x = 0 and another value, such as x = 1, to create a system of equations.
1. When x = 0:
f(0) = 5(0)3 + 7(0) + 5 = 5
2. When x = 1:
f(1) = 5(1)3 + 7(1) + 5 = 5 + 7 + 5 = 17
Next, we substitute these values into the remainder polynomial Ax + B:
– For x = 0:
B = 5
– For x = 1:
A(1) + B = 17
A + 5 = 17
A = 12
Now we have determined that A = 12 and B = 5. Therefore, the remainder is:
12x + 5
In conclusion, the remainder when 5x3 + 7x + 5 is divided by x2 is 12x + 5.