Finding The Difference Of Polynomials: A Step-by-Step Guide

by Alex Johnson 60 views

Understanding how to find the difference between polynomials is a fundamental skill in algebra. This article will guide you through the process, using the example of finding the difference between the polynomials (10m + 0) and (7m + 4). We'll break down each step, explain the concepts involved, and help you master this essential mathematical operation.

Understanding Polynomials

Before we dive into the specific problem, let's briefly review what polynomials are. A polynomial is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents. For instance, 10m + 0 and 7m + 4 are both polynomials. The variable here is m, and the coefficients are the numbers multiplying m (10 and 7, respectively) and the constants (0 and 4).

Key Components of Polynomials

To effectively work with polynomials, it’s crucial to understand their components:

  • Variables: These are symbols (usually letters like x, y, or m) that represent unknown values. In our case, m is the variable.
  • Coefficients: These are the numbers that multiply the variables. In 10m + 0, the coefficient of m is 10, and in 7m + 4, it's 7.
  • Constants: These are terms without variables. In 10m + 0, the constant is 0, and in 7m + 4, it's 4.
  • Terms: These are the individual parts of a polynomial, separated by addition or subtraction. For example, in 10m + 0, 10m and 0 are the terms.

Knowing these components helps in performing operations like addition, subtraction, multiplication, and division on polynomials.

Setting Up the Subtraction

The question asks us to find the difference between the polynomials (10m + 0) and (7m + 4). This means we need to subtract the second polynomial from the first. Mathematically, this can be written as:

(10m + 0) - (7m + 4)

The key to subtracting polynomials is to correctly distribute the negative sign. When subtracting one polynomial from another, you are essentially adding the negative of the second polynomial to the first. This is a critical step, and overlooking it can lead to errors.

Distributing the Negative Sign

To distribute the negative sign, we multiply each term inside the second polynomial by -1. So, -(7m + 4) becomes -7m - 4. Our expression now looks like this:

(10m + 0) - 7m - 4

This step is crucial because it transforms the subtraction problem into an addition problem, making it easier to combine like terms.

Combining Like Terms

Now that we have distributed the negative sign, the next step is to combine like terms. Like terms are terms that have the same variable raised to the same power. In our expression (10m + 0) - 7m - 4, the like terms are 10m and -7m, and the constants 0 and -4.

Identifying Like Terms

Before combining, it's essential to correctly identify like terms. Remember:

  • Terms with the same variable and exponent are like terms (e.g., 3x^2 and -5x^2).
  • Constants are like terms (e.g., 5 and -2).
  • Terms with different variables or exponents are not like terms (e.g., 2x and 3y, or 4x^2 and 2x).

Combining Like Terms in Our Example

In our expression, we have:

  • 10m and -7m are like terms.
  • 0 and -4 are like terms.

To combine them, we add their coefficients:

  • 10m + (-7m) = 3m
  • 0 + (-4) = -4

So, the simplified expression is:

3m - 4

Evaluating the Given Options

Now, let's look at the given options and see which one correctly represents the difference we found, which is 3m - 4. The options are:

A. |10m + (-7m)| + |(-5) + 4| B. (10m + 7m) + |(-0) + (-4)| C. |(-10m) + (-7m)| + (0 + 4) D. |10m + (-7m)| + |0 + (-4)|

We need to simplify each option and see which one matches our result, 3m - 4.

Analyzing Option A: |10m + (-7m)| + |(-0) + (-4)|

Let's simplify this option step by step:

  1. |10m + (-7m)| = |3m|
  2. |(-0) + (-4)| = |-4| = 4
  3. So, the expression becomes |3m| + 4. This is not equivalent to 3m - 4 because of the absolute value and the addition sign.

Analyzing Option B: (10m + 7m) + |(-0) + (-4)|

Simplify this option:

  1. (10m + 7m) = 17m
  2. |(-0) + (-4)| = |-4| = 4
  3. So, the expression becomes 17m + 4. This is also not equivalent to 3m - 4.

Analyzing Option C: |(-10m) + (-7m)| + (0 + 4)

Simplify this option:

  1. |(-10m) + (-7m)| = |-17m|
  2. (0 + 4) = 4
  3. So, the expression becomes |-17m| + 4. This is not equivalent to 3m - 4 due to the absolute value and the addition sign.

Analyzing Option D: |10m + (-7m)| + |0 + (-4)|

Simplify this option:

  1. |10m + (-7m)| = |3m|
  2. |0 + (-4)| = |-4| = 4
  3. So, the expression becomes |3m| + |-4| = |3m| + 4. This is also not equivalent to 3m - 4 because of the absolute value.

Correct Representation of the Difference

None of the provided options directly match our simplified expression 3m - 4. However, let's re-evaluate how we can express the difference using absolute values to see if we can find an equivalent representation.

The difference between two polynomials can be thought of as the absolute value of the difference, but we need to ensure the signs are correctly handled. Our simplified difference is 3m - 4. We need an expression that captures both the magnitude and the sign.

Let's reconsider option D: |10m + (-7m)| + |0 + (-4)|. We simplified this to |3m| + 4. While this isn't exactly 3m - 4, it gives us a clue. We need to account for the negative sign in front of the 4.

The Correct Approach

The key is to realize that we need to express the difference in a way that maintains the correct sign. The correct representation should be:

|10m + (-7m)| + |0 + (-4)| simplifies to |3m| + |-4| which equals |3m| + 4.

However, none of the options perfectly represents 3m - 4. The closest option, D, gives us the magnitude but not the correct sign for the constant term. Therefore, it seems there might be an issue with the provided options.

Conclusion

Finding the difference between polynomials involves distributing the negative sign, combining like terms, and simplifying the expression. In the case of (10m + 0) - (7m + 4), the simplified form is 3m - 4. After evaluating the given options, we found that none of them perfectly match this result, primarily due to the presence of absolute values that don't correctly account for the negative sign in the constant term. It's essential to carefully follow each step and double-check your work to ensure accuracy in polynomial subtraction. Explore more about polynomials and algebraic expressions on trusted educational websites such as Khan Academy Algebra.