Simplifying Polynomials Finding The First Term In Standard Form

Have you ever felt like you're wrestling with a mathematical beast, trying to tame a wild polynomial? Well, Raj faced just such a challenge! He was given the task of simplifying a polynomial and putting it into that oh-so-satisfying standard form. Let's dive into the problem and see how Raj tackled it, and maybe even learn a thing or two along the way, guys.

The Polynomial Puzzle

The polynomial Raj was asked to simplify was:

$2 x^2 y+8 x^3-x y^2-2 x^3+3 x y^2+6 y^3$

It looks a bit like a jumbled mess, right? Our mission, should we choose to accept it, is to help Raj simplify this expression and arrange it in standard form. But what is standard form, you ask? Good question! In simple terms, standard form means arranging the terms in descending order of their degree. The degree of a term is the sum of the exponents of its variables. For example, the degree of $x^2y$ is 2 + 1 = 3, while the degree of $x^3$ is simply 3. Constant terms, like $6y^3$ (which Raj correctly identified!), have a degree determined by the variable part. So, the degree of $6y^3$ is 3.

Before we even think about the first term, let’s focus on simplifying the polynomial. This involves combining like terms – those terms that have the same variables raised to the same powers. It's like sorting socks in your laundry; you group the ones that match! In our polynomial, we have like terms involving $x^2y$, $x^3$, and $xy^2$. Let’s gather them together:

• $x^2y$ terms: $2x^2y$
• $x^3$ terms: $8x^3$ and $-2x^3$
• $xy^2$ terms: $-xy^2$ and $3xy^2$
• $y^3$ terms: $6y^3$ (this lone wolf is already in its own category!)

Now, let's combine these like terms. Remember, we're essentially adding or subtracting the coefficients (the numbers in front of the variables) while keeping the variable part the same. Think of it like this: 8 apples minus 2 apples equals 6 apples. So, $8x^3 - 2x^3$ becomes $6x^3$.

Simplifying Step-by-Step

Let's apply this to our polynomial:

1. Combine the $x^3$ terms: $8x^3 - 2x^3 = 6x^3$
2. Combine the $xy^2$ terms: $-xy^2 + 3xy^2 = 2xy^2$

Now, let’s rewrite the polynomial with the combined terms:

$2x^2y + 6x^3 + 2xy^2 + 6y^3$

We're getting closer! But remember, standard form is the name of the game. So, we need to arrange these terms in descending order of their degree. Let's figure out the degree of each term:

• $2x^2y$: Degree is 2 + 1 = 3
• $6x^3$: Degree is 3
• $2xy^2$: Degree is 1 + 2 = 3
• $6y^3$: Degree is 3

Uh oh! We have a bunch of terms with the same degree (3). What do we do now? Well, when terms have the same degree, we usually arrange them in lexicographical order based on the variables. This essentially means we look at the exponents of x first. The term with the highest exponent of x comes first. If the exponents of x are the same, we then look at the exponents of y, and so on.

So, let's rearrange our simplified polynomial:

1. $6x^3$ (degree 3, highest exponent of x)
2. $2x^2y$ (degree 3, next highest exponent of x)
3. $2xy^2$ (degree 3, lower exponent of x)
4. $6y^3$ (degree 3, no x)

Therefore, the simplified polynomial in standard form is:

$6x^3 + 2x^2y + 2xy^2 + 6y^3$

Unveiling the First Term

And now, for the grand reveal! Raj simplified the polynomial and correctly identified the final term as $6y^3$. But what is the first term of the polynomial in standard form? Drumroll, please…

The first term is $\bf{6x^3}$.

So, there you have it! We've not only simplified the polynomial and put it into standard form, but we've also uncovered the first term. Raj would be proud!

Diving Deeper into Polynomial Simplification

Simplifying polynomials is a fundamental skill in algebra, and it's one that you'll use time and time again in more advanced math courses. Let's break down the key concepts we've touched upon and explore some additional tips and tricks to master this skill, alright?

Understanding Like Terms is Crucial

The heart of polynomial simplification lies in the ability to identify and combine like terms. As we discussed earlier, like terms are those that have the same variables raised to the same powers. Think of it as grouping similar objects together. You wouldn't add apples and oranges, would you? Similarly, you can't directly combine terms like $x^2$ and $x^3$ because they represent different quantities.

Why are like terms so important? Because they allow us to reduce the complexity of a polynomial. By combining like terms, we can express the polynomial in its most concise form, making it easier to work with and understand. This is especially critical when solving equations or performing other algebraic operations.

How do you confidently identify like terms? Here's a checklist to keep in mind:

1. Same Variables: The terms must have the same variables (e.g., both have x and y).
2. Same Exponents: The corresponding variables must have the same exponents (e.g., both have $x^2$ and $y$).

For example, $3x^2y$ and $-5x^2y$ are like terms because they both have x raised to the power of 2 and y raised to the power of 1. However, $3x^2y$ and $3xy^2$ are not like terms because the exponents of x and y are different.

Mastering the Art of Combining Like Terms

Once you've identified like terms, the next step is to combine them. This involves adding or subtracting their coefficients. Remember, the coefficient is the numerical factor that multiplies the variable part of the term. It's the number that sits in front of the variables.

The Golden Rule: When combining like terms, you only add or subtract the coefficients. The variable part remains the same. It's like saying 3 apples + 2 apples = 5 apples. The