Simplify Polynomials: Combining Like Terms

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Hey guys! Let's dive into the world of polynomials and learn how to simplify them like pros. Polynomials might sound intimidating, but trust me, they're not as scary as they seem. In this article, we're going to break down the process of simplifying polynomials by combining like terms and arranging them in descending order. We'll use a specific example to illustrate each step, so you can follow along and master this essential skill.

Understanding Polynomials: The Building Blocks

Before we jump into simplifying, let's make sure we're all on the same page about what polynomials actually are. Polynomials are algebraic expressions that consist of variables and coefficients, combined using addition, subtraction, and non-negative integer exponents. Think of them as mathematical Lego sets, where variables are the building blocks and coefficients are the connectors.

For example, the expression -4p^2 - 3p - 2 is a polynomial. It has three terms: -4p^2, -3p, and -2. The coefficients are -4, -3, and -2, respectively, and the variable is p. The exponents are 2 (for the first term) and 1 (for the second term, since p is the same as p^1). The last term, -2, is called a constant term because it doesn't have a variable.

Polynomials can have one or more terms. A polynomial with one term is called a monomial (e.g., 5x), a polynomial with two terms is called a binomial (e.g., 2x + 3), and a polynomial with three terms is called a trinomial (e.g., x^2 - 4x + 7). Polynomials with more than three terms are simply called polynomials.

The degree of a term in a polynomial is the exponent of the variable in that term. For example, the degree of the term -4p^2 is 2, and the degree of the term -3p is 1. The degree of a constant term is always 0 (since we can think of -2 as -2p^0).

The degree of the polynomial itself is the highest degree of any of its terms. In the example -4p^2 - 3p - 2, the highest degree is 2, so the degree of the polynomial is 2. This means it's a quadratic polynomial.

Now that we've covered the basics, let's move on to the star of the show: simplifying polynomials.

Identifying Like Terms: Spotting the Twins

The key to simplifying polynomials is to combine like terms. But what exactly are like terms? Like terms are terms that have the same variable raised to the same exponent. They're like twins in the polynomial world – they look alike and can be combined without changing their fundamental identity.

For example, 3x^2 and -5x^2 are like terms because they both have the variable x raised to the power of 2. Similarly, 7p and -2p are like terms because they both have the variable p raised to the power of 1. However, 2x^2 and 2x are not like terms because they have the same variable but different exponents.

Constant terms are also considered like terms because they can be combined directly. For instance, 5 and -8 are like terms.

Identifying like terms is like playing a matching game. You need to carefully examine each term and look for its twin – a term with the same variable and exponent.

In our example problem, (-4p^2 - 3p - 2) - (-p^2 - 4), we have the following terms: -4p^2, -3p, -2, -p^2, and -4. Let's identify the like terms:

  • -4p^2 and -p^2 are like terms (both have p^2).
  • -3p has no like terms in this expression.
  • -2 and -4 are like terms (both are constants).

Now that we've identified the like terms, we can move on to the next step: combining them.

Combining Like Terms: The Art of Addition and Subtraction

Once you've identified the like terms, combining them is a breeze. To combine like terms, you simply add or subtract their coefficients while keeping the variable and exponent the same. It's like adding apples to apples or bananas to bananas – you can only combine things that are the same.

For example, to combine 3x^2 and -5x^2, we add their coefficients (3 + (-5) = -2) and keep the variable and exponent the same, resulting in -2x^2.

Similarly, to combine 7p and -2p, we add their coefficients (7 + (-2) = 5) and keep the variable the same, resulting in 5p.

Constant terms are combined by simply adding or subtracting them as regular numbers. For example, 5 + (-8) = -3.

Now, let's apply this to our example problem: (-4p^2 - 3p - 2) - (-p^2 - 4). The first thing we need to do is distribute the negative sign in the second parentheses. Remember, subtracting a quantity is the same as adding its opposite. So, we can rewrite the expression as:

-4p^2 - 3p - 2 + p^2 + 4

Now we can identify and combine the like terms:

  • Combine the p^2 terms: -4p^2 + p^2 = -3p^2
  • The -3p term has no like terms, so it remains as -3p.
  • Combine the constant terms: -2 + 4 = 2

So, after combining like terms, our expression becomes:

-3p^2 - 3p + 2

We're almost there! The final step is to arrange the polynomial in descending order.

Arranging in Descending Order: Putting Things in Their Place

Arranging a polynomial in descending order means writing the terms from the highest degree to the lowest degree. It's like lining up students by height, with the tallest student first and the shortest student last.

In our example, the polynomial is -3p^2 - 3p + 2. Let's identify the degrees of each term:

  • -3p^2 has a degree of 2.
  • -3p has a degree of 1.
  • 2 has a degree of 0 (since it's a constant term).

To arrange the polynomial in descending order, we write the term with the highest degree first, followed by the term with the next highest degree, and so on. So, the polynomial in descending order is:

-3p^2 - 3p + 2

In this case, the polynomial was already in descending order, but that won't always be the case. Sometimes you'll need to rearrange the terms to get them in the correct order.

Putting It All Together: The Complete Solution

Let's recap the steps we took to simplify the polynomial (-4p^2 - 3p - 2) - (-p^2 - 4):

  1. Distribute the negative sign: -4p^2 - 3p - 2 + p^2 + 4
  2. Identify like terms:
    • -4p^2 and p^2
    • -3p (no like terms)
    • -2 and 4
  3. Combine like terms: -3p^2 - 3p + 2
  4. Arrange in descending order: -3p^2 - 3p + 2 (already in descending order)

So, the simplified form of the polynomial is -3p^2 - 3p + 2.

Why Simplify Polynomials? The Real-World Connection

You might be wondering,