Polynomials: Combine Like Terms & Standard Form

Jan 26, 2026 by Andrew McMorgan 48 views

Hey math whizzes and algebra adventurers! Today, we're diving headfirst into the awesome world of polynomials. You know, those cool expressions with variables, coefficients, and exponents? We're going to tackle a common task that often trips people up: combining like terms and putting a polynomial into standard form. It sounds a bit technical, but trust me, guys, it's totally manageable once you get the hang of it. We'll break down this specific problem step-by-step, so by the end, you’ll be a pro at simplifying these algebraic beasts.

Understanding Polynomials and Standard Form

First things first, let's get our definitions straight. A polynomial is basically a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. Think of terms like $3x^2$ , $-5y$ , or $7$ . When we talk about combining like terms, we mean simplifying an expression by adding or subtracting terms that have the same variable raised to the same power. For instance, $2x$ and $5x$ are like terms because they both have $x$ to the power of 1, so we can combine them to get $7x$ . However, $3x^2$ and $4x$ are not like terms because the powers of $x$ are different (2 and 1, respectively). You can't just smoosh them together willy-nilly!

Now, standard form for a polynomial is all about order. It means arranging the terms from the highest power of the variable down to the lowest power (the constant term). So, if you have terms like $5x^3$ , $2x$ , and $-7x^2$ , the standard form would be $5x^3 - 7x^2 + 2x$ . The term with the highest exponent comes first, followed by the next highest, and so on, until you reach the term with no variable (the constant). This organized way of writing polynomials makes them easier to compare, add, subtract, and work with in more complex operations. It’s like tidying up your algebraic workspace so you can see everything clearly!

The Problem at Hand: Simplifying Our Polynomial

Alright, let's get down to business with the polynomial given: $rac{1}{3} x+8 x^4- rac{2}{3} x^2-x$ . Our mission, should we choose to accept it (and we totally should!), is to combine any like terms and then arrange the whole thing in standard form. This means we need to find terms that have the same variable and exponent and either add or subtract their coefficients. Then, we'll line them up from highest power to lowest.

Looking at our polynomial, we have the following terms:

$8x^4$ (a term with $x$ to the power of 4)
$- rac{2}{3}x^2$ (a term with $x$ to the power of 2)
$rac{1}{3}x$ (a term with $x$ to the power of 1)
$-x$ (another term with $x$ to the power of 1)

Do you see any like terms here? Yep, you got it! The terms $rac{1}{3}x$ and $-x$ are like terms because they both involve $x$ raised to the power of 1. The other terms, $8x^4$ and $- rac{2}{3}x^2$ , are unique in their powers ( $x^4$ and $x^2$ , respectively), so they can't be combined with anything else.

Step 1: Combining Like Terms

Let's focus on combining our like terms: $rac{1}{3}x$ and $-x$ . Remember, $-x$ is the same as $-1x$ . So, we need to calculate $rac{1}{3} - 1$ . To do this, we need a common denominator. Since 1 can be written as $rac{3}{3}$ , we have $rac{1}{3} - rac{3}{3}$ . Subtracting the numerators gives us $rac{1-3}{3} = - rac{2}{3}$ .

So, $rac{1}{3}x - x$ simplifies to $- rac{2}{3}x$ .

Now, let's put our simplified polynomial back together. We had the terms $8x^4$ , $- rac{2}{3}x^2$ , and the combined term $- rac{2}{3}x$ . Our polynomial now looks like this: $8x^4 - rac{2}{3}x^2 - rac{2}{3}x$ .

Step 2: Arranging in Standard Form

Our next job is to put this simplified polynomial into standard form. Remember, standard form means writing the terms in descending order of their exponents. Let's look at the exponents in our current polynomial: $8x^4$ , $- rac{2}{3}x^2$ , $- rac{2}{3}x$ .

The exponents are 4, 2, and 1.

Arranging these from highest to lowest, we get:

Exponent 4: $8x^4$
Exponent 2: $- rac{2}{3}x^2$
Exponent 1: $- rac{2}{3}x$

Putting it all together in this order, we get: $8x^4 - rac{2}{3}x^2 - rac{2}{3}x$ .

Comparing with the Options

Now, let's compare our final answer with the options provided:

A. $8 x^4- rac{1}{3} x^2-x$ B. $8 x^4+ rac{2}{3} x^2+ rac{2}{3} x$ C. $8 x^4- rac{2}{3} x^2- rac{2}{3} x$

Boom! Our result matches option C exactly. It looks like we correctly combined the like terms ( $rac{1}{3}x$ and $-x$ to get $- rac{2}{3}x$ ) and then arranged the resulting terms ( $8x^4$ , $- rac{2}{3}x^2$ , and $- rac{2}{3}x$ ) in descending order of exponents.

Why Other Options Are Incorrect

Let's quickly see why options A and B didn't make the cut. For option A, $8 x^4- rac{1}{3} x^2-x$ , the like terms weren't combined correctly. It seems like the calculation for the $x$ terms might have been off, or perhaps the $x^2$ term was mistakenly altered. Remember, $rac{1}{3}x - x = - rac{2}{3}x$ , not $- rac{1}{3}x^2 - x$ . The powers are also different, so combining $rac{1}{3}x$ and $-x$ should result in an $x$ term, not an $x^2$ term. This option also fails to combine the original $x$ terms correctly, leaving them as $-x$ and not combining them with $rac{1}{3}x$ .

Option B, $8 x^4+ rac{2}{3} x^2+ rac{2}{3} x$ , has incorrect signs and coefficients for the $x^2$ and $x$ terms. Our original polynomial had $- rac{2}{3}x^2$ and the combination of $rac{1}{3}x$ and $-x$ resulted in $- rac{2}{3}x$ . Option B seems to have changed the sign of the $x^2$ term to positive and also changed the sign of the combined $x$ term to positive, which is not what happened when we performed the calculations based on the given polynomial. It's super important to pay close attention to those plus and minus signs, guys!

Key Takeaways for Polynomial Mastery

So, what did we learn from this algebraic escapade? Firstly, always identify like terms – those with the identical variable(s) raised to the identical power(s). Secondly, when combining them, perform the arithmetic operations on their coefficients. Thirdly, remember that standard form is the tidy way to write polynomials, ordered from the highest exponent down to the lowest. Keep these steps in mind, and you'll conquer any polynomial simplification task that comes your way. It’s all about attention to detail and practicing those core math skills. Keep practicing, and you'll be writing polynomials in standard form like a boss!

This whole process might seem a bit tedious at first, but the more you practice, the quicker and more intuitive it becomes. Think of it like learning a new dance move; at first, you're concentrating on every step, but soon, it flows naturally. Polynomials are fundamental in algebra and beyond, appearing in everything from physics equations to computer graphics. So, mastering these basics is a massive step in your math journey. Don't shy away from challenges; embrace them as opportunities to grow your understanding and your problem-solving prowess. Keep that mathematical curiosity alive, and you’ll always find something new and exciting to learn.

We've successfully navigated the process of combining like terms and arranging our polynomial in standard form. The key was carefully identifying the terms with the same variable and exponent, performing the subtraction for the $x$ terms, and then ordering the resulting terms from highest power to lowest. This methodical approach ensures accuracy and leads us directly to the correct answer. Remember, in mathematics, precision is key, and following the rules consistently will always lead you to the right solution. Keep up the great work, and happy solving!