When a polynomial P(x) is divided by x+2, the quotient is x^2 - 4x + 1 and the remainder is 8, what is the polynomial?
Well, well, well, looks like we've got ourselves a polynomial puzzle! Let's solve it together, shall we?
So, if the polynomial P(x) is divided by x+2, we're given that the quotient is x^2 - 4x + 1 and the remainder is 8. How can we figure out what our polynomial P(x) is?
First, let's set up the equation:
P(x) = (x+2)(x^2 - 4x + 1) + 8
Now, let's simplify and expand:
P(x) = x^3 - 2x^2 - 4x^2 + 8x + x - 8 + 8
P(x) = x^3 - 6x^2 + 9x
Voila! The polynomial P(x) is simply x^3 - 6x^2 + 9x. Isn't mathematics a-MAIZE-ing?
To determine the polynomial when P(x) is divided by x+2, we can set up the division as follows:
x^2 - 4x + 1
x+2 | P(x)
Since the remainder is given as 8, we can write:
P(x) = (x+2)(x^2 - 4x + 1) + 8
Expanding the product on the right side:
P(x) = x^3 - 4x^2 + x + 2x^2 - 8x + 2 + 8
Combining like terms:
P(x) = x^3 - 2x^2 - 7x + 10
Therefore, the polynomial P(x) is x^3 - 2x^2 - 7x + 10.
To find the polynomial, we need to reverse the process of polynomial division.
Given the quotient x^2 - 4x + 1 and the remainder 8, we can write the polynomial P(x) as:
P(x) = (x+2)(x^2 - 4x + 1) + remainder
Now, let's expand the product (x+2)(x^2 - 4x + 1):
P(x) = (x^2 - 4x + 1)(x+2) + 8
Multiplying the terms, we get:
P(x) = x^3 + 2x^2 - 4x^2 - 8x + x + 2 - 8
Then, combining like terms:
P(x) = x^3 - 2x^2 - 7x - 6
So, the polynomial P(x) is x^3 - 2x^2 - 7x - 6.
P(x) / (x+2) = x^2 - 4x + 1 + 8/(x+2)
P(x) = (x+2)(x^2 - 4x + 1) + 8
= x^3 - 2x^2 - 7x + 2 + 8
= x^3 - 2x^2 - 7x + 10