How to Subtract Polynomials: Easy Algebra 1 Method

📚 Understanding Polynomials

Polynomials are algebraic expressions containing variables and coefficients, combined using addition, subtraction, and multiplication. Subtracting them involves combining like terms after distributing the negative sign.

🗓️ A Brief History

The concept of polynomials dates back to ancient civilizations, with early forms appearing in Babylonian and Greek mathematics. However, the systematic study and manipulation of polynomials as we know it developed over centuries, with significant contributions from mathematicians in India, the Islamic world, and Europe.

🔑 Key Principles of Subtracting Polynomials

• 🔍 Identify Like Terms: Like terms have the same variable raised to the same power (e.g., $3x^2$ and $-5x^2$).
• ➖ Distribute the Negative Sign: When subtracting one polynomial from another, distribute the negative sign to each term in the second polynomial.
• ➕ Combine Like Terms: Add or subtract the coefficients of the like terms.
• ✏️ Simplify: Write the resulting polynomial in standard form (decreasing order of exponents).

📝 Step-by-Step Method

1. Write the Polynomials: Write down the polynomial you are subtracting *from* first, followed by a minus sign and then the polynomial you are subtracting. Enclose the second polynomial in parentheses: $(4x^2 + 3x - 2) - (x^2 - 5x + 1)$
2. Distribute the Negative Sign: Change the sign of each term inside the parentheses: $4x^2 + 3x - 2 - x^2 + 5x - 1$
3. Combine Like Terms: Group and combine like terms: $(4x^2 - x^2) + (3x + 5x) + (-2 - 1)$
4. Simplify: Simplify each group: $3x^2 + 8x - 3$

💡 Example 1: Subtracting Two Simple Polynomials

Let's subtract $(2x + 3)$ from $(5x - 2)$.

$(5x - 2) - (2x + 3) = 5x - 2 - 2x - 3 = (5x - 2x) + (-2 - 3) = 3x - 5$

➗ Example 2: Subtracting Polynomials with Higher Degrees

Subtract $(3x^2 - 2x + 1)$ from $(7x^2 + 5x - 4)$.

$(7x^2 + 5x - 4) - (3x^2 - 2x + 1) = 7x^2 + 5x - 4 - 3x^2 + 2x - 1 = (7x^2 - 3x^2) + (5x + 2x) + (-4 - 1) = 4x^2 + 7x - 5$

🧮 Example 3: Subtracting Polynomials with Multiple Variables

Subtract $(2x^2y - xy + 3y^2)$ from $(5x^2y + 4xy - y^2)$.

$(5x^2y + 4xy - y^2) - (2x^2y - xy + 3y^2) = 5x^2y + 4xy - y^2 - 2x^2y + xy - 3y^2 = (5x^2y - 2x^2y) + (4xy + xy) + (-y^2 - 3y^2) = 3x^2y + 5xy - 4y^2$

✏️ Practice Quiz

1. Subtract $(x + 2)$ from $(3x + 5)$.
2. Subtract $(2x - 1)$ from $(4x + 3)$.
3. Subtract $(x^2 + 3x - 2)$ from $(2x^2 - x + 1)$.
4. Subtract $(3x^2 - 2x + 4)$ from $(5x^2 + x - 3)$.
5. Subtract $(x^3 + 2x^2 - x)$ from $(3x^3 - x^2 + 2x)$.
6. Subtract $(2x^3 - x^2 + 3)$ from $(4x^3 + 2x^2 - 1)$.
7. Subtract $(4xy + y^2 - x^2)$ from $(6xy - 2y^2 + 3x^2)$.

✅ Solutions to the Practice Quiz

1. $2x + 3$
2. $2x + 4$
3. $x^2 - 4x + 3$
4. $2x^2 + 3x - 7$
5. $2x^3 - 3x^2 + 3x$
6. $2x^3 + 3x^2 - 4$
7. $2xy - 3y^2 + 4x^2$

🌍 Real-world Applications

• 💰 Finance: Calculating profit or loss by subtracting expenses from revenue.
• 📏 Engineering: Determining the difference in measurements or dimensions.
• 🌡️ Science: Finding the change in temperature or other variables in experiments.

🔑 Key Takeaways

• 👍 Distribute Carefully: Ensure you correctly distribute the negative sign to each term in the polynomial being subtracted.
• 🧮 Combine Like Terms: Only combine terms with the same variable and exponent.
• 🧐 Check Your Work: Double-check your calculations to avoid errors with signs.

🎓 Conclusion

Subtracting polynomials might seem tricky at first, but with practice and a clear understanding of the steps involved, it can become a straightforward process. Remember to distribute the negative sign carefully and combine like terms accurately.