What is the remainder when $2x^3 - 10x^2 + 6x + 2$ is divided by $x - 2$? (A) -66 (B) -10 (C) -$x^3 + 5x^2 - 3x - 1$ (D) $x^3 - 5x^2 + 3x + 1$ - HSC - SSCE Mathematics Extension 1 - Question 2 - 2016 - Paper 1

To find the remainder when dividing a polynomial by a linear factor, we can use the Remainder Theorem. This theorem states that the remainder of the polynomial $f(x)$ when divided by $x - c$ is equal to $f(c)$.

In this case, we have:

$f(x) = 2x^3 - 10x^2 + 6x + 2$

We want to evaluate this polynomial at $c = 2$:

$f(2) = 2(2)^3 - 10(2)^2 + 6(2) + 2$

Calculating each term gives:

• $2(8) = 16$
• $10(4) = 40$
• $6(2) = 12$
• The constant term is $2$

Putting it all together:

$f(2) = 16 - 40 + 12 + 2$

Now, simplify:

$f(2) = 16 + 12 + 2 - 40 = 30 - 40 = -10$

Thus, the remainder when $2x^3 - 10x^2 + 6x + 2$ is divided by $x - 2$ is -10.