Find the remainder when $6x^4-14x^3-4x^2+2x-26$ is divided by $2x - 6.$
I know that I can use the remainder theorem to find the remainder without actually doing the long division. The remainder theorem says that the remainder is equal to the value of the polynomial when x is replaced by the number that makes the divisor zero. So I need to find the value of x that makes $2x - 6 = 0.$ I can do that by adding 6 to both sides and dividing by 2. I get $x = 3.$ So the remainder is the same as $6(3)^4-14(3)^3-4(3)^2+2(3)-26.$ I get $6(81)-14(27)-4(9)+6-26.$ This is $486-378-36+6-26.$ This is $52.$ 