Question

when a polynomial x^3-ax^2-2x+2a+6 leaves a remainder (a+2) when divided by (x-a), find the value of a.​

Answer 1

`a = 4`

Explanation:

We are given that the polynomial:

`\displaystyle x^3 - ax^2 - 2x + 2a + 6`

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):

`\displaystyle P(a) = a+2`

Substitute:

`\displaystyle (a)^3 - a(a)^2 -2(a) + 2a + 6 = a + 2`

Simplify and solve for a:

`\displaystyle \begin{aligned} a^3 - a^3 - 2a + 2a + 6 &= a + 2 \\ \\ 6 &= a+2 \\ \\ a &= 4\end{aligned}`

In conclusion, the value of a is 4.