Adding Subtracting Polynomials Answer Key

Mastering Polynomial Addition and Subtraction: A Comprehensive Guide with Answer Key

Polynomials are fundamental algebraic expressions that form the building blocks for more advanced mathematical concepts. Understanding how to add and subtract polynomials is crucial for success in algebra and beyond. This comprehensive guide will walk you through the process, providing clear explanations, examples, and a detailed answer key to solidify your understanding. We'll cover everything from basic concepts to more complex scenarios, ensuring you're equipped to tackle any polynomial addition and subtraction problem with confidence.

Understanding Polynomials: A Quick Refresher

Before diving into addition and subtraction, let's review the basics of polynomials. A polynomial is an expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. Each part of a polynomial separated by a plus or minus sign is called a term. A term consists of a coefficient (a number) and a variable (or variables) raised to a power. The highest power of the variable in a polynomial is called its degree.

For example, consider the polynomial 3x² + 5x - 7.

• Terms: 3x², 5x, and -7
• Coefficients: 3, 5, and -7
• Variables: x
• Degree: 2 (because the highest power of x is 2)

A polynomial with one term is called a monomial, a polynomial with two terms is a binomial, and a polynomial with three terms is a trinomial. Polynomials with more than three terms are simply called polynomials.

Adding Polynomials: A Step-by-Step Approach

Adding polynomials is straightforward. The key is to combine like terms. Like terms are terms that have the same variables raised to the same powers. You can think of it as combining apples with apples and oranges with oranges.

Steps:

1. Identify like terms: Carefully examine the polynomials and identify terms with the same variables raised to the same powers.
2. Group like terms: Arrange the polynomial so that like terms are grouped together. You can use parentheses to help organize this step.
3. Combine like terms: Add the coefficients of the like terms. Remember that the variable and its exponent remain unchanged.
4. Simplify: Write the resulting polynomial in descending order of powers (from highest to lowest).

Example 1:

Add (3x² + 5x - 7) + (2x² - 3x + 4)

1. Like terms: 3x² and 2x², 5x and -3x, -7 and 4
2. Grouping: (3x² + 2x²) + (5x - 3x) + (-7 + 4)
3. Combining: 5x² + 2x - 3
4. Simplified: 5x² + 2x - 3

Example 2 (with multiple variables):

Add (4xy² + 2x²y - 3y) + (x²y + 5xy² + 2y)

1. Like terms: 4xy² and 5xy², 2x²y and x²y, -3y and 2y
2. Grouping: (4xy² + 5xy²) + (2x²y + x²y) + (-3y + 2y)
3. Combining: 9xy² + 3x²y - y
4. Simplified: 3x²y + 9xy² - y

Subtracting Polynomials: A Slightly Different Approach

Subtracting polynomials is similar to addition, but with one crucial difference: you need to distribute the negative sign to each term in the second polynomial before combining like terms. This is essentially changing the sign of every term in the polynomial being subtracted.

Steps:

1. Rewrite as addition: Rewrite the subtraction problem as an addition problem by changing the subtraction sign to addition and changing the sign of every term in the second polynomial.
2. Identify like terms: Identify terms with the same variables raised to the same powers.
3. Group like terms: Arrange the polynomial so that like terms are grouped together.
4. Combine like terms: Add the coefficients of the like terms.
5. Simplify: Write the resulting polynomial in descending order of powers.

Example 1:

Subtract (5x² - 2x + 6) - (3x² + 4x - 1)

1. Rewrite as addition: (5x² - 2x + 6) + (-3x² - 4x + 1)
2. Like terms: 5x² and -3x², -2x and -4x, 6 and 1
3. Grouping: (5x² - 3x²) + (-2x - 4x) + (6 + 1)
4. Combining: 2x² - 6x + 7
5. Simplified: 2x² - 6x + 7

Example 2 (with multiple variables):

Subtract (7a³b² - 3ab + 2b²) - (2a³b² + 5ab - b²)

1. Rewrite as addition: (7a³b² - 3ab + 2b²) + (-2a³b² - 5ab + b²)
2. Like terms: 7a³b² and -2a³b², -3ab and -5ab, 2b² and b²
3. Grouping: (7a³b² - 2a³b²) + (-3ab - 5ab) + (2b² + b²)
4. Combining: 5a³b² - 8ab + 3b²
5. Simplified: 5a³b² - 8ab + 3b²

Adding and Subtracting Polynomials: Advanced Scenarios

Let's tackle some more complex scenarios to further solidify your understanding.

Example 3: Adding and Subtracting Multiple Polynomials

Simplify: (2x³ + 4x² - x + 3) + (x³ - 2x² + 5x - 1) - (3x³ + x² - 2x + 4)

First, distribute the negative sign to the third polynomial:

(2x³ + 4x² - x + 3) + (x³ - 2x² + 5x - 1) + (-3x³ - x² + 2x - 4)

Now, group and combine like terms:

(2x³ + x³ - 3x³) + (4x² - 2x² - x²) + (-x + 5x + 2x) + (3 - 1 - 4)

Simplify: 0x³ + x² + 6x - 2 = x² + 6x - 2

Example 4: Polynomials with Higher Degrees

Add: (5x⁴ - 2x³ + 3x² - x + 7) + (x⁴ + 4x³ - 2x² + 5x - 3)

Group and combine like terms:

(5x⁴ + x⁴) + (-2x³ + 4x³) + (3x² - 2x²) + (-x + 5x) + (7 - 3)

Simplify: 6x⁴ + 2x³ + x² + 4x + 4

Frequently Asked Questions (FAQ)

• Q: What happens if I don't have like terms? A: If you don't have like terms, you simply write the terms as they are, in descending order of powers. You can't combine unlike terms.

• Q: Can I subtract polynomials vertically? A: Yes! You can arrange the polynomials vertically, aligning like terms, and then subtract column by column. Remember to distribute the negative sign to each term in the second polynomial.

• Q: What if I have a polynomial with a missing term? A: It's helpful to represent missing terms with a coefficient of zero. This makes combining like terms easier. For example, if you have x² + 5, you can think of it as x² + 0x + 5.

• Q: How do I check my answer? A: A great way to check your answer is to substitute a value for the variable (like x=1 or x=2) in both the original expression and your simplified result. If they give the same numerical result, you've likely simplified correctly.

Conclusion: Mastering Polynomials

Adding and subtracting polynomials is a foundational skill in algebra. By understanding the concepts of like terms, grouping, and combining coefficients, you can confidently tackle any polynomial addition and subtraction problem. Remember to practice regularly, use the steps outlined in this guide, and don't hesitate to refer back to the examples and FAQ section for additional support. With consistent effort, you'll master this skill and be well-prepared for more advanced algebraic concepts. Keep practicing, and soon you'll find polynomial operations as easy as adding and subtracting numbers!