Adding and Subtracting Polynomials

When working with polynomials, one of the fundamental operations you'll encounter is the addition and subtraction of these expressions. This article will explore the techniques involved in adding and subtracting polynomials with clear explanations, practical examples, and some practice problems for you to try out.

What are Polynomials?

Before diving into the operations, let's briefly recall what polynomials are. A polynomial is an expression consisting of variables (often represented by letters) and coefficients, combined using addition, subtraction, and multiplication. The general form of a polynomial in one variable \(x\) can be expressed as:

\[ a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 \]

where:

  • \(a_n, a_{n-1}, \ldots, a_0\) are coefficients (which can be any real numbers),
  • \(n\) is a non-negative integer indicating the degree of the polynomial, and
  • \(x\) is the variable.

Adding Polynomials

To add polynomials, you combine like terms. Like terms are terms that contain the same variable(s) raised to the same power. Here’s a step-by-step approach to adding polynomials:

Step 1: Identify like terms

When you look at the polynomials, identify the terms that have the same variable and exponent.

Step 2: Combine like terms

For the terms identified, add or subtract their coefficients while keeping the variable part unchanged.

Example of Adding Polynomials:

Let’s consider the polynomials \( P(x) = 4x^3 + 3x^2 - 2x + 5 \) and \( Q(x) = 2x^3 - x^2 + 4x - 7 \).

Step 1: Identify like terms

  • The like terms are:
    • \(4x^3\) and \(2x^3\)
    • \(3x^2\) and \(-x^2\)
    • \(-2x\) and \(4x\)
    • \(5\) and \(-7\)

Step 2: Combine like terms \[ P(x) + Q(x) = (4x^3 + 2x^3) + (3x^2 - x^2) + (-2x + 4x) + (5 - 7) \] Calculating each group:

  • For \(x^3\) terms: \(4 + 2 = 6\) gives \(6x^3\)
  • For \(x^2\) terms: \(3 - 1 = 2\) gives \(2x^2\)
  • For \(x\) terms: \(-2 + 4 = 2\) gives \(2x\)
  • For the constant terms: \(5 - 7 = -2\) gives \(-2\)

Putting it all together: \[ P(x) + Q(x) = 6x^3 + 2x^2 + 2x - 2 \]

Practice Problem 1:

Add the following polynomials: \[ A(x) = 5x^4 + 3x^3 - 2x + 6 \quad \text{and} \quad B(x) = x^4 - 4x^3 + 5x - 3 \] Solution: Identify like terms and combine them.

Subtracting Polynomials

Subtracting polynomials is similar to adding them, but instead of adding the coefficients of like terms, you will subtract them.

Step 1: Write the second polynomial in a suitable form

Change \(Q(x)\) to \(-Q(x)\) by negating each term.

Step 2: Combine like terms

This is done in the same way as addition, just with the new polynomial.

Example of Subtracting Polynomials:

Using the same polynomials as before \( P(x) \) and \( Q(x) \): \[ P(x) = 4x^3 + 3x^2 - 2x + 5 \ Q(x) = 2x^3 - x^2 + 4x - 7 \] To subtract \( Q(x) \) from \( P(x) \):

Step 1: Negate \(Q(x)\) \[ -Q(x) = -2x^3 + x^2 - 4x + 7 \]

Step 2: Combine like terms \[ P(x) - Q(x) = P(x) + (-Q(x)) = (4x^3 + 3x^2 - 2x + 5) + (-2x^3 + x^2 - 4x + 7) \] Calculating:

  • For \(x^3\) terms: \(4 - 2 = 2\) gives \(2x^3\)
  • For \(x^2\) terms: \(3 + 1 = 4\) gives \(4x^2\)
  • For \(x\) terms: \(-2 - 4 = -6\) gives \(-6x\)
  • For the constant terms: \(5 + 7 = 12\) gives \(12\)

Putting it all together: \[ P(x) - Q(x) = 2x^3 + 4x^2 - 6x + 12 \]

Practice Problem 2:

Subtract the following polynomials: \[ C(x) = 7x^5 + 3x^2 + x - 4 \quad \text{and} \quad D(x) = 2x^5 - x^2 + 2x + 5 \] Solution: Negate \(D(x)\) and combine like terms.

More Practice Problems

As you continue practicing adding and subtracting polynomials, try these problems:

  1. Add: \[ E(x) = 3x^2 + 5x - 1 \quad \text{and} \quad F(x) = 4x^2 - 2x + 8 \]

  2. Subtract: \[ G(x) = 6x^4 + 2x^3 - x + 9 \quad \text{and} \quad H(x) = x^4 + 3x^3 - 7x + 5 \]

Solutions:

  1. Solution for adding \(E(x)\) and \(F(x)\):

    • Combine the like terms to get your answer.

  2. Solution for subtracting \(G(x)\) from \(H(x)\):

    • Negate terms in \(H(x)\) and combine appropriately.

Conclusion

Understanding how to add and subtract polynomials is essential for mastering algebra. By following the steps outlined in this article and practicing with the problems provided, you will build a strong foundation in working with polynomial expressions. Remember, the key is to focus on combining like terms effectively and always keeping track of the signs of each term. Happy calculating!