Evaluate n³ + 5n² + 5n + 2 when n = 2

Alright guys, let’s dive into evaluating this expression! We’ve got n³ + 5n² + 5n + 2 , and we need to find its value when n = 2 . Sounds like a plan? Let’s break it down step by step so it’s super clear and easy to follow. Grab your pencils (or keyboards!) and let’s get started!

Step-by-Step Evaluation

First, we’re given the expression:

n³ + 5n² + 5n + 2

And we know that:

n = 2

So, we’re going to substitute n with 2 in the expression. This means everywhere we see n , we’ll replace it with 2 . Here’s how it looks:

(2)³ + 5(2)² + 5(2) + 2

Now, let’s calculate each term individually. This will make it easier to keep track of everything.

Calculating 2³

The term 2³ means 2 * 2 * 2 . So, let’s do the math:

2 * 2 = 4

4 * 2 = 8

So, 2³ = 8 .

Calculating 5(2)²

Next, we need to calculate 5(2)² . Remember, the order of operations (PEMDAS/BODMAS) tells us to handle the exponent first. So, we calculate 2² first:

2² = 2 * 2 = 4

Now, we multiply this result by 5:

5 * 4 = 20

So, 5(2)² = 20 .

Calculating 5(2)

This one is straightforward. We simply multiply 5 by 2:

5 * 2 = 10

So, 5(2) = 10 .

Putting It All Together

Now that we’ve calculated each term, let’s put them all back into the expression:

8 + 20 + 10 + 2

Now, we just need to add these numbers together:

8 + 20 = 28

28 + 10 = 38

38 + 2 = 40

So, the final result is 40 .

Conclusion

Therefore, when n = 2 , the value of the expression n³ + 5n² + 5n + 2 is 40 . Wasn’t that fun? We took a seemingly complex expression, broke it down into smaller, manageable parts, and solved it step by step. You rock!

Deep Dive into Polynomial Evaluation

Alright, let’s get a bit more in-depth, shall we? Understanding how to evaluate polynomials isn’t just about plugging in numbers; it’s about grasping the underlying structure and principles. We’re going to explore why this process works, the broader applications, and some tricks to make it even easier. Buckle up!

Understanding Polynomials

First, let’s make sure we’re all on the same page. A polynomial is an expression consisting of variables (like n ), coefficients (like 5 ), and exponents (like ³ ), combined using addition, subtraction, and multiplication. The expression we worked with, n³ + 5n² + 5n + 2 , is a perfect example.

Components of a Polynomial

• Variable: This is the unknown value, represented by a letter (in our case, n ).
• Coefficient: The number multiplied by the variable (e.g., 5 in 5n² ).
• Exponent: The power to which the variable is raised (e.g., 3 in n³ ).
• Constant: A term without a variable (e.g., 2 in our expression).

Why Substitution Works

When we substitute a value for n , we’re essentially asking: “What is the value of this expression if n is this specific number?” By replacing n with its given value, we transform the expression from a symbolic representation to a numerical one. This allows us to perform arithmetic operations and arrive at a single, definitive value.

Order of Operations

Remember PEMDAS/BODMAS? It’s crucial! It dictates the order in which we perform operations:

1. P arentheses / B rackets
2. E xponents / O rders
3. M ultiplication and D ivision (from left to right)
4. A ddition and S ubtraction (from left to right)

Following this order ensures we get the correct result every time.

Applications of Polynomial Evaluation

Evaluating polynomials isn’t just an abstract math exercise; it has real-world applications in various fields.

Engineering

Engineers use polynomials to model systems and predict behavior. For example, they might use a polynomial to describe the trajectory of a projectile or the stress on a bridge.

Computer Science

In computer graphics, polynomials are used to create curves and surfaces. Evaluating these polynomials allows computers to render complex shapes and animations.

Economics

Economists use polynomials to model economic trends and make predictions about future performance.

Physics

Polynomials appear in many physics equations, describing everything from motion to energy.

Tricks for Easier Evaluation

Okay, let’s talk about making this process even smoother. Here are a few tricks to keep in mind:

Simplify First

Before substituting, see if you can simplify the expression. Combining like terms or factoring can make the evaluation easier.

Break It Down

As we did earlier, break the expression into smaller parts. Calculate each term individually before adding them together. This reduces the chance of making mistakes.

Use Technology

Don’t be afraid to use a calculator or software to help with the calculations, especially for more complex expressions. Tools like Wolfram Alpha can be incredibly useful.

Practice, Practice, Practice

The more you practice, the faster and more accurate you’ll become. Try evaluating different polynomials with various values of n .

Common Mistakes to Avoid

Even with a solid understanding, it’s easy to make mistakes. Here are a few common pitfalls to watch out for:

Incorrect Order of Operations

Always follow PEMDAS/BODMAS. It’s the golden rule!

Sign Errors

Be careful with negative signs. Make sure you’re applying them correctly.

Arithmetic Errors

Double-check your calculations. Simple arithmetic errors can throw off the entire result.

Incorrect Substitution

Ensure you’re substituting the value of n correctly in all instances.

Advanced Techniques

For those who want to take it a step further, let’s touch on some advanced techniques.

Horner’s Method

Horner’s method is an efficient algorithm for evaluating polynomials. It reduces the number of multiplications needed, making it faster for computers to calculate.

Polynomial Remainder Theorem

The Polynomial Remainder Theorem states that if you divide a polynomial f(n) by n - a , the remainder is f(a) . This can be useful for finding the value of a polynomial at a specific point.

Real-World Examples and Applications

To really drive the point home, let’s look at some real-world examples where polynomial evaluation comes into play. Trust me, it’s more relevant than you might think!

Example 1: Trajectory of a Projectile

Imagine you’re launching a rocket. The height h of the rocket at time t can be modeled by a polynomial equation like:

h(t) = -4.9t² + 25t + 2

Here, -4.9t² represents the effect of gravity, 25t represents the initial upward velocity, and 2 is the initial height. If you want to know the height of the rocket at, say, t = 3 seconds, you would evaluate h(3) :

h(3) = -4.9(3)² + 25(3) + 2

h(3) = -4.9(9) + 75 + 2

h(3) = -44.1 + 75 + 2

h(3) = 32.9

So, at t = 3 seconds, the rocket is approximately 32.9 meters high.

Example 2: Modeling Growth

Let’s say you’re tracking the growth of a bacteria colony. The number of bacteria B after x hours can be modeled by:

B(x) = 2x³ + 3x² + x + 100

If you want to know how many bacteria there will be after x = 5 hours, you evaluate B(5) :

B(5) = 2(5)³ + 3(5)² + 5 + 100

B(5) = 2(125) + 3(25) + 5 + 100

B(5) = 250 + 75 + 5 + 100

B(5) = 430

So, after 5 hours, there will be 430 bacteria.

Example 3: Cost Function

In business, cost functions are often represented by polynomials. For example, the cost C of producing q units of a product might be:

C(q) = 0.1q² + 5q + 1000

Here, 0.1q² represents variable costs, 5q represents the cost of materials, and 1000 is the fixed cost. If you want to know the cost of producing q = 50 units, you evaluate C(50) :

C(50) = 0.1(50)² + 5(50) + 1000

C(50) = 0.1(2500) + 250 + 1000

C(50) = 250 + 250 + 1000

C(50) = 1500

So, the cost of producing 50 units is $1500.

The Importance of Precision

In these real-world examples, precision is key. A small error in calculation can lead to significant discrepancies in the final result. For instance, in the rocket trajectory example, an incorrect calculation could lead to misjudging the landing point. In the bacteria growth example, it could affect the accuracy of predictions. And in the cost function example, it could impact budgeting and pricing decisions.

Final Thoughts

Polynomial evaluation is a fundamental skill with far-reaching applications. By mastering this skill, you’re not just learning math; you’re equipping yourself with a tool that can be applied in various fields, from engineering to economics. So keep practicing, keep exploring, and keep pushing your boundaries! You’ve got this!