To determine the result of dividing the polynomial px = x^4 + 2x^3 + 3x^2 + ax + 3a + 7 by x + 1, we will use polynomial long division or synthetic division.
First, we can apply the Remainder Theorem, which states that the remainder of a polynomial f(x) when divided by x – c is simply f(c). Here, since we are dividing by x + 1, we will substitute -1 into the polynomial.
Calculating p(-1):
p(-1) = (-1)^4 + 2(-1)^3 + 3(-1)^2 + a(-1) + 3a + 7
p(-1) = 1 – 2 + 3 – a + 3a + 7
p(-1) = 1 – 2 + 3 + 2a + 7 = 9 + 2a
The remainder when dividing px by x + 1 is 9 + 2a.
This means that the expression can be written as:
px = (x + 1)Q(x) + (9 + 2a),
where Q(x) is the quotient obtained from the division. Thus, the result of dividing the polynomial by x + 1 yields a quotient and a remainder of 9 + 2a.