Let's go through another example of an algebraic expression, showing the step-by-step process to simplify and evaluate it.

Example:

Expression: $4x^3 - 2x^2 + 7x - 5$

This expression contains four terms: $4x^3$, $-2x^2$, $7x$, and $-5$. Let's simplify it and then evaluate it for a specific value of $x$.

Step 1: Simplifying the Expression

First, check if the expression can be simplified by combining like terms. In this case, all terms are distinct (they have different powers of $x$), so the expression is already in its simplest form: $4x^3 - 2x^2 + 7x - 5$

Step 2: Evaluating the Expression for a Specific Value of $x$

Next, let's evaluate the expression for $x = 2$. To do this, substitute $x = 2$ into the expression: $4(2)^3 - 2(2)^2 + 7(2) - 5$

Step 3: Performing the Calculations

Now, we'll calculate each term step by step:

1. Calculate the exponent and multiply by the coefficient: $2^3 = 8 \quad \text{and} \quad 4 \times 8 = 32$ $2^2 = 4 \quad \text{and} \quad -2 \times 4 = -8$ $7 \times 2 = 14$ $\text{The constant term is} \, -5$

2. Substitute these results back into the expression: $32 - 8 + 14 - 5$

Step 4: Simplify the Expression

Finally, perform the addition and subtraction: $32 - 8 = 24$ $24 + 14 = 38$ $38 - 5 = 33$

Final Answer:

The value of the expression $4x^3 - 2x^2 + 7x - 5$ when $x = 2$ is $33$.

Would you like more details on any step?

Here are 8 related questions to deepen your understanding:

1. How would the expression change if the coefficients were different?
2. What happens if $x = -2$ in the above expression?
3. Can you identify the degree of the expression?
4. How would you factor this expression if it were factorable?
5. What is the significance of the constant term in the expression?
6. How would you evaluate the expression for a fractional value of $x$?
7. Can you graph this expression as a function of $x$?
8. What are like terms, and why can't we combine terms in the original expression?

Tip: When simplifying expressions, always look for like terms first, as combining them can make evaluation simpler!