91Ó°ÊÓ

Function expansion is an essential step when working with expressions, especially in calculus. Here, we aim to simplify expressions for better understanding and to carry forward operations. In our example, we have the function: \[ f(x) = 3x^2 + 2 \] When we substitute \( x + h \) into the function, we need to expand: \[ f(x+h) = 3(x+h)^2 + 2 \] This requires distributing the terms within the parentheses and applying algebraic rules. The expansion of \( (x+h)^2 \): \[ (x+h)^2 = x^2 + 2xh + h^2 \] incorporating it back into our original function: \[ 3(x^2 + 2xh + h^2) + 2 = 3x^2 + 6xh + 3h^2 + 2 \] This process lays the foundation for further simplification. Proper expansion ensures correctness in subsequent steps.

Simplifying expressions makes them easier to handle. In our example, after expanding the function, we get: \[ f(x+h) = 3x^2 + 6xh + 3h^2 + 2 \] We need to substitute this back into the difference quotient: \[ \frac{(3x^2 + 6xh + 3h^2 + 2) - (3x^2 + 2)}{h} \] Next, we simplify the numerator by combining like terms. Notice that \( 3x^2 + 2 \) in\ our expansion cancels out with \( 3x^2 + 2 \) from the original function: \[ 3x^2 + 6xh + 3h^2 + 2 - 3x^2 - 2 = 6xh + 3h^2 \] Now we have a simpler numerator that we can work with.

Factoring is the process of splitting an expression into a product of simpler terms. This step can often help with canceling out terms for more straightforward simplification. For the equation: \[ \frac{6xh + 3h^2}{h} \] We factor out \( h \) from the numerator: \[ \frac{h(6x + 3h)}{h} \] Now, we cancel \( h \) in both the numerator and the denominator. This leaves us with: \[ 6x + 3h \] Thus, the simplified form of the difference quotient is: \[ 6x + 3h \] Always remember, factoring and canceling common terms simplifies expressions and makes them easier to interpret and solve.