What is the remainder when $2x^3 - 10x^2 + 6x + 2$ is divided by $x - 2$? - HSC - SSCE Mathematics Extension 1 - Question 2 - 2016 - Paper 1

To find the remainder of the polynomial $P(x) = 2x^3 - 10x^2 + 6x + 2$ when divided by $x - 2$, we can use the Remainder Theorem which states that the remainder of the division of a polynomial by a linear divisor can be found by evaluating the polynomial at the root of the divisor.

In this case, the root of $x - 2$ is $x = 2$. We need to calculate $P(2)$:

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

Calculating each term:

• $2(2)^3 = 2 \times 8 = 16$
• $-10(2)^2 = -10 \times 4 = -40$
• $6(2) = 12$
• $2 = 2$

Now, sum these values:

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

Calculating step-by-step:

• $16 - 40 = -24$
• $-24 + 12 = -12$
• $-12 + 2 = -10$

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