Use long division to find the quetient and the remainder for the following division: \( \frac{6 x^{3}-11 x^{2}+25 x-23}{3 x-4} \) a.) Quotient \( 2 x^{2}-x+7 \) Remainder. 5 b.) Quotient \( 2 x^{2}+x+7 \) Remainder. 5 c.) Quotient \( 2 x^{2}+x+7 \) Remainder. -5 (

To perform long division on \( \frac{6x^3 - 11x^2 + 25x - 23}{3x - 4} \), we start by dividing the leading term \( 6x^3 \) by \( 3x \), which gives us \( 2x^2 \). Multiplying \( 2x^2 \) by \( 3x - 4 \) gives \( 6x^3 - 8x^2 \). Subtracting this from the original polynomial leads to \( -3x^2 + 25x - 23 \). Next, we divide \( -3x^2 \) by \( 3x \) to get \( -x \) and repeat the process. Carrying on this way, we discover that the quotient is indeed \( 2x^2 + x + 7 \) with a remainder of \( -5 \). Now, if you've got a craving for some math fun, think about how long division helps illuminate many seemingly complex mathematical problems! It’s like peeling the layers off an onion—each quotient and remainder brings us closer to the answer. And remember, with practice, this method becomes as easy and satisfying as pie. Want to dive deeper? Check out books that explore polynomial long division, diving into advanced algebra or even calculus, pushing your math skills to new heights!