Adding Polynomials: A Step-by-Step Guide

Hey guys! Today, we're diving into the awesome world of polynomials and tackling a super common question: "What is the sum of the polynomials?" Specifically, we'll be working through the example: $\left(1.3 t^3+0.4 t^2-24 t\right)+\left(8-18 t+0.6 t^2\right)$. Don't worry if this looks a little intimidating at first; by the end of this article, you'll be a pro at adding these bad boys together. We'll break it down piece by piece, making sure you understand each step. So, grab your notebooks, and let's get started on this mathematical adventure!

Understanding Polynomials and Addition

Before we jump into solving our specific problem, let's quickly recap what polynomials are and what it means to add them. Polynomials are basically mathematical expressions consisting of variables (like 't' in our case) and coefficients, which are combined using addition, subtraction, and multiplication, but not division by a variable. They can have one or more terms, and the highest power of the variable in any term is called the degree of the polynomial. For instance, in $1.3 t^3+0.4 t^2-24 t$, the variable is 't', the coefficients are 1.3, 0.4, and -24, and the powers are 3, 2, and 1. The highest power is 3, so it's a third-degree polynomial.

When we talk about adding polynomials, we're essentially combining like terms. Like terms are terms that have the same variable raised to the same power. Think of them as buddies – you can only add or subtract buddies of the same kind. For example, you can add $3t^2$ and $5t^2$ to get $8t^2$, but you can't directly combine $3t^2$ and $2t$ because they have different powers of 't'. So, the core strategy for adding polynomials is to identify these like terms and sum their coefficients.

Our problem involves adding two polynomials: $(1.3 t^3+0.4 t^2-24 t)$ and $(8-18 t+0.6 t^2)$. Notice that the second polynomial is written in a slightly different order, and it's missing a $t^3$ term (which is equivalent to $0t^3$). This is totally normal, and it's why organizing your terms is super important. We want to line up the terms with the same powers of 't' to make addition a breeze. So, the first step in our problem is to rewrite the second polynomial so that its terms are in descending order of power, just like the first one. This would give us $0.6 t^2 - 18t + 8$. Now, we're ready to combine them!

Step-by-Step Polynomial Addition

Alright, let's get down to business with our specific problem: $\left(1.3 t^3+0.4 t^2-24 t\right)+\left(8-18 t+0.6 t^2\right)$. The first crucial step is to remove the parentheses. Since we are adding the two polynomials, the signs of the terms inside the second set of parentheses remain unchanged. So, we can rewrite the expression as: $1.3 t^3+0.4 t^2-24 t+0.6 t^2-18 t+8$. Now that the parentheses are gone, the next big move is to group the like terms together. This makes it much easier to see which terms can be combined. Let's identify them:

• t³ terms: We only have one $t^3$ term, which is $1.3 t^3$.
• t² terms: We have $0.4 t^2$ and $0.6 t^2$. These are like terms!
• t terms: We have $-24t$ and $-18t$. These are also like terms.
• Constant terms: We have the number $8$. This is our only constant term.

Now, let's rearrange the expression to have the like terms next to each other. This is a great organizational trick that will prevent mistakes. So, we get: $1.3 t^3 + (0.4 t^2 + 0.6 t^2) + (-24 t - 18 t) + 8$. You can see how grouping them makes it super clear what we need to add.

The final step in this part is to combine the coefficients of the like terms. This is where the actual addition happens.

• For the $t^3$ terms: We have $1.3 t^3$. Since there's nothing else to add it to, it stays as is.
• For the $t^2$ terms: We add the coefficients $0.4$ and $0.6$. So, $0.4 + 0.6 = 1.0$. This gives us $1.0 t^2$, or simply $t^2$.
• For the $t$ terms: We add $-24$ and $-18$. Remember, adding a negative number is the same as subtracting. So, $-24 - 18 = -42$. This gives us $-42t$.
• For the constant terms: We have $8$. It remains $8$.

Putting it all together, our sum is $1.3 t^3 + 1.0 t^2 - 42 t + 8$. We can write $1.0 t^2$ as just $t^2$ for a cleaner look. So, the final answer is $1.3 t^3 + t^2 - 42 t + 8$. Easy peasy, right?

Comparing with the Options

Now that we've calculated the sum of the polynomials, let's look at the options provided to see which one matches our result. Our calculated sum is $1.3 t^3 + t^2 - 42 t + 8$. Let's examine each option:

• A. $1.3 t^3+t^2-42 t+8$: This option matches our calculated sum exactly! We have the $1.3 t^3$ term, the $t^2$ term (which is $1t^2$), the $-42t$ term, and the constant $+8$. This looks like our winner, guys!

• B. $1.3 t^3+t^2-6 t+8$: This option is close, but it's different in the 't' term. Our calculation resulted in $-42t$, while this option has $-6t$. This means the coefficients of the 't' terms were not combined correctly in this option, or there was a mistake in the initial grouping.

• C. $1.9 t^2-42 t+8$: This option is missing the $t^3$ term entirely. It also seems to have combined the $t^2$ terms incorrectly (perhaps adding $1.3t^3$ and $0.6t^2$ by mistake, which is not allowed as they are not like terms). Our calculation clearly showed a $1.3 t^3$ term in the result.

• D. $1.9 t^3+8.4 t^2-42 t$: This option also seems to have made mistakes. The $t^3$ term is $1.9t^3$, which doesn't match our $1.3t^3$. The $t^2$ term is $8.4t^2$, which is also incorrect; we found $1.0t^2$. Furthermore, this option is missing the constant term $+8$. This option has several errors compared to our correct calculation.

Based on our detailed step-by-step calculation, Option A is the correct answer. It perfectly reflects the sum obtained by correctly combining like terms from the given polynomials. It's always a good idea to double-check your work, especially when dealing with negative numbers and decimals, to ensure you haven't made any silly mistakes. This process of adding polynomials is fundamental in algebra, and mastering it will open doors to more complex mathematical concepts.

Why Polynomial Addition Matters

So, why do we even bother learning how to add polynomials, you ask? Polynomials are the building blocks for a huge chunk of mathematics, especially in algebra. They appear everywhere, from describing curves and shapes in geometry to modeling real-world phenomena in physics and economics. For instance, the path of a projectile can often be described by a quadratic polynomial, and understanding how to manipulate these polynomials helps us predict where the projectile will land or how high it will go.

In calculus, which is the study of change, polynomials are fundamental. Derivatives and integrals of polynomial functions are straightforward to calculate, and they form the basis for understanding more complex functions. When you're solving equations or systems of equations, you'll often encounter polynomials. Being able to simplify expressions by adding or subtracting polynomials is a crucial skill that makes these more complex tasks manageable. Think of it like learning your ABCs before you can write a novel; polynomial addition is a foundational skill.

Moreover, understanding how to combine like terms in polynomials builds a strong foundation for working with algebraic expressions in general. It teaches you the importance of organization, attention to detail, and logical step-by-step problem-solving. These are skills that extend far beyond mathematics and are valuable in any academic or professional pursuit. So, next time you're crunching numbers and adding polynomials, remember you're not just solving a math problem; you're honing essential critical thinking skills that will serve you well throughout your life. Keep practicing, and you'll become a polynomial pro in no time!

Conclusion

We've successfully navigated the process of adding two polynomials, $\left(1.3 t^3+0.4 t^2-24 t\right)$ and $\left(8-18 t+0.6 t^2\right)$. By carefully removing parentheses, grouping like terms, and combining their coefficients, we arrived at the sum $1.3 t^3 + t^2 - 42 t + 8$. This matches Option A, proving it to be the correct answer. Remember, the key to adding polynomials is to treat each power of the variable as a distinct category and only combine terms that belong to the same category. This methodical approach ensures accuracy and builds confidence. Keep practicing these types of problems, guys, and you'll find that polynomial operations become second nature. Happy calculating!