Use synthetic division and the remainder theorem to find P(a).
p(x)=x^3+4x^2-3x+2 a=2
To find P(a) using synthetic division and the remainder theorem, we will divide the polynomial p(x) = x^3 + 4x^2 - 3x + 2 by x - a = x - 2.
The coefficients of p(x) are 1, 4, -3, and 2. We will set up the synthetic division as follows:
2 | 1 4 -3 2
--------------------
2 12 18
--------------------
1 6 15 20
The numbers at the bottom row are the coefficients of the quotient polynomial, in order from highest power to lowest power. The last number, 20, is the remainder.
Therefore, the quotient polynomial is: x^2 + 6x + 15.
To find P(a), we substitute a = 2 into the quotient polynomial:
P(2) = 2^2 + 6(2) + 15
= 4 + 12 + 15
= 31.
Hence, P(2) = 31.