Answer:
[tex]a = 4[/tex]
Step-by-step explanation:
We are given that the polynomial:
[tex]\displaystyle x^3 - ax^2 - 2x + 2a + 6[/tex]
Has the remainder (a + 2) when divided by (x - a), and we want to determine the value of a.
Recall that from the Polynomial Remainder Theorem, when dividing a polynomial P(x) by a binomial (x - a), the remainder will be given by P(a).
Since the remainder is (a + 2) when divided by (x - a), P(a) must equal (a + 2):
[tex]\displaystyle P(a) = a+2[/tex]
Substitute:
[tex]\displaystyle (a)^3 - a(a)^2 -2(a) + 2a + 6 = a + 2[/tex]
Simplify and solve for a:
[tex]\displaystyle \begin{aligned} a^3 - a^3 - 2a + 2a + 6 &= a + 2 \\ \\ 6 &= a+2 \\ \\ a &= 4\end{aligned}[/tex]
In conclusion, the value of a is 4.
