Evaluate $-x^3+3x+4$ When $x=-1$: A Step-by-Step Guide

Hey guys! Let's dive into a math problem together. Today, we're going to tackle the expression $-x^3 + 3x + 4$ and figure out what it equals when $x$ is $-1$. This might seem a bit daunting at first, but trust me, we'll break it down step by step, so it's super easy to understand. So, grab your pencils, and let's get started!

Understanding the Expression

Before we jump into plugging in numbers, let's make sure we understand what the expression $-x^3 + 3x + 4$ really means. Think of it as a set of instructions. We have a variable, $x$, which is just a placeholder for a number. In this case, we're told that $x$ is equal to $-1$. The expression tells us to do a few things:

1. First, we need to cube $x$ (that's $x$ multiplied by itself three times: $x * x * x$).
2. Then, we multiply that result by $-1$.
3. Next, we multiply $x$ by $3$.
4. Finally, we add all those results together, along with the number $4$.

It's like a recipe, right? We just need to follow the steps in the right order. Understanding this breakdown is crucial because it sets the stage for accurate substitution and evaluation. We need to be mindful of the order of operations (PEMDAS/BODMAS), which dictates that we handle exponents before multiplication and addition. This initial understanding helps prevent common mistakes and makes the entire process smoother. So, with a clear grasp of the expression, we're now ready to substitute the value of $x$ and move towards finding the solution. Let's keep going!

Step 1: Substitute $x$ with $-1$

The first thing we need to do is replace every $x$ in the expression with the value $-1$. This is called substitution. So, our expression $-x^3 + 3x + 4$ becomes:

$-(-1)^3 + 3(-1) + 4$

See how we just swapped out the $x$'s with $-1$? Make sure you put the $-1$ in parentheses. This is super important, especially when we're dealing with negative numbers and exponents. The parentheses help us keep track of the negative sign and make sure we're applying the exponent correctly. This simple step of correct substitution is fundamental to getting the right answer. Messing up the signs or missing parentheses can throw off the whole calculation. So, always double-check that you've substituted correctly before moving on to the next step. Now that we've made the substitution, the expression is ready for the next operation, which involves dealing with the exponent. Let's tackle that next!

Step 2: Evaluate the Exponent

Now, let's deal with the exponent. We have $(-1)^3$, which means $-1$ multiplied by itself three times: $(-1) * (-1) * (-1)$.

• $-1 * -1 = 1$ (a negative times a negative is a positive)
• $1 * -1 = -1$ (a positive times a negative is a negative)

So, $(-1)^3 = -1$.

This is a key step, guys! Remember that a negative number raised to an odd power will always be negative, while a negative number raised to an even power will be positive. This rule is something you'll use a lot in math, so it's good to get it down. Exponents can sometimes be tricky, especially when negative signs are involved. Taking the time to carefully evaluate the exponent ensures that we carry the correct value forward into the rest of the calculation. A mistake here can ripple through the entire problem, leading to a wrong final answer. So, we've successfully calculated the exponent, and we're one step closer to solving the expression. What’s next? We'll use this result in the following steps to simplify the rest of the equation. Let's move on!

Step 3: Multiply

Next up, we've got some multiplication to do. Our expression now looks like this:

$-(-1) + 3(-1) + 4$

We have two multiplications to handle:

• $-(-1)$: A negative of a negative is a positive, so $-(-1) = 1$.
• $3(-1)$: A positive times a negative is a negative, so $3(-1) = -3$.

Multiplication is a fundamental arithmetic operation, and it's crucial to get it right. Here, we're dealing with the multiplication of integers, including negative numbers, so paying close attention to the signs is essential. Remember the rules: a negative times a negative yields a positive, and a positive times a negative (or vice versa) results in a negative. Applying these rules correctly is what allows us to accurately simplify the expression. We've successfully handled the multiplication parts, and now the expression is becoming simpler and easier to manage. Let’s keep going! With these multiplications out of the way, we’re now ready to tackle the final operation: addition.

Step 4: Add

Now, let's add the numbers together. Our expression has been simplified to:

$1 + (-3) + 4$

We can add these numbers from left to right:

• $1 + (-3) = -2$
• $-2 + 4 = 2$

So, the final result is $2$.

Addition is the final step in this evaluation, and it brings all our previous work together to reach the solution. It's like the last piece of the puzzle fitting into place. When adding integers, especially when negative numbers are involved, it's helpful to think about it as moving along a number line. Adding a positive number moves you to the right, while adding a negative number moves you to the left. This mental image can make it easier to visualize the process and avoid errors. We’ve carefully combined the terms, and we've arrived at our final answer. This demonstrates the power of breaking down a complex problem into manageable steps. With the addition completed, we've successfully evaluated the expression!

Conclusion

So, when we evaluate the expression $-x^3 + 3x + 4$ when $x = -1$, we get $2$. Great job, guys! You've successfully navigated through the steps of substitution, exponentiation, multiplication, and addition. Remember, the key to solving these kinds of problems is to break them down into smaller, manageable steps and to pay close attention to the order of operations. Keep practicing, and you'll become a math whiz in no time! Understanding how to approach and solve these expressions is a valuable skill that you’ll use throughout your math journey. Keep up the great work, and let’s tackle the next math challenge! You got this!