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Operations with Polynomials

This guide covers the fundamental operations with polynomials: addition, subtraction, and multiplication. Learn the importance of combining like terms, ensuring the polynomials are similar when performing operations. You'll find step-by-step examples demonstrating how to add and subtract polynomials effectively, including the distribution of negative signs in subtraction. Additionally, understand how to multiply polynomials by adding exponents when combining variables. Engage in practice exercises with a partner to reinforce your learning and ensure accurate understanding of these concepts.

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Operations with Polynomials

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  1. Operations with Polynomials Just the basics

  2. Addition and subtraction • When adding or subtracting polynomials remember they must be exactly the same!!! • 5x + 3x = 8x That is okay • 5x2 + 3x ≠ 8x2 These terms are not the same!!!!

  3. Addition • Example 1 • (5x2 – 2x + 3) + (3x2 – 4x2 + 7) Combine like terms within parentheses 1st • (5x2 – 2x + 3) + (-x2+ 7) Drop the parentheses and combine like terms. • 4x2 – 2x + 10 • Example 2 • (3x2 + 5x – 2x5) + (6x3 – x2 + 11) Anything the same inside? Nope! Drop the parentheses and combine like terms. • -2x5 + 6x3 + 2x2 + 5x + 11

  4. Subtraction • Pretty much the same, but before you drop the parentheses you have to distribute the negative sign. (Subtraction sign and negative are interchangeable!) • Example 1 • (3x2 – 2x + 1) – (x2 + 5x – 9) Distribute the negative 1st. • 3x2– 2x + 1– x2 - 5x + 9 Combine like terms! • 2x2 – 7x + 10 • Example 2 • (-7x2 – x + 5) - (2 – x3 + 3x) Distribute. • -7x2 – x + 5 – 2 + x3 – 3x Combine like terms • x3 – 7x2 – 4x + 3 Traditionally, the terms would be written highest power to lowest. I do not really care!!!

  5. Multiplication • Only rule to remember is that when you multiple variables, you add the exponents. • Example 1 Example 2 • (-5x3)(9x2) = -45x5 5x2(3x5 – x3 + 7x – 4)= • 15x7 – 5x5 + 35x3 -20x2 • Example 3 • (2x – 5)(x2 – 2x + 3) Distribute in two steps. • 2x3 – 4x2 + 6x -5x2 +10x -15 Combine like terms. • 2x3 – 9x2 + 16x - 15

  6. Try on your own • Find a partner. • Get a worksheet. • Take turns working out the problems on the worksheet. • Support your partner on their turn. • Check your answers when you are done. • If you are right, get your homework. If not, find another group that can help you!!!

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