Question 3 Use long division to divide \( 2 x^{4}-3 x^{3}-24 x^{2}+76 x-60 \) by \( 2 x-3 \). What is the new expression? \( x^{3}+2 x^{2}-6 x+20 \) \( -x^{3}+6 x+20 \) \( 2 x^{3}-x^{2}-12 x+20 \) \( x^{3}-12 x+20 \)

When you carry out long division of the polynomial \( 2x^{4}-3x^{3}-24x^{2}+76x-60 \) by \( 2x-3 \), you are essentially trying to simplify the polynomial into a more manageable form. Long division in algebra is very similar to the long division of numbers, where you divide, multiply, subtract, and bring down the next term. The result will yield a new polynomial expression and a possible remainder. Upon completing the long division, the answer you get is a polynomial \( x^{3}+2x^{2}-6x+20 \). This means that when you divide your initial polynomial by \( 2x-3 \), you end up with a simpler expression where you can further analyze or manipulate the components of the polynomial for additional calculations or graphing. This method is extremely helpful in polynomial functions and higher algebraic operations!