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10.2 – Adding and Subtracting Polynomials

Key Terms

  • Area – The size of a surface. It is measured in square units.
  • Collecting Like Terms – Simplifying an expression by grouping together terms that have the same variables and powers and then adding or subtracting these terms.
  • Tiles – Tools that can help add, subtract, multiply, and divide some polynomials.
    • Small squares represent 1. Small rectangles represent x. Larger squares represent x^{2}.

Review

Methods
  • Techniques are methods.  In other words, they are “ways of doing something.”
Area
  • Area is a two-dimensional measurement.
  • Area\;=\; x\bullet x=x^{2}
  • That’s why we cannot use the tiles to represent degrees higher than 2.

Alg1B 10.2 Tile Area

Operations
  • Sum: addition
  • Difference: subtraction
  • Product: multiplication
  • Quotient: division.

Notes

Adding Polynomials Using Tiles
  • Tiles are labeled using their area (length times width)

Alg1B 10.2 Tiles1

  • How to Use Tiles to Add Polynomials
    • Step 1: Use tiles to show each polynomial.
    • Step 2: Combine all the tiles.
    • Step 3: Write a polynomial for the combined tiles.

Alg1B 10.2 Tiles2


  • Binomial Example: (3x+2)+(2x+1)Alg1B 10.2 Tiles Sum

Answer: 5x+3

  • Trinomial Example

Alg1B 10.2 Tiles Sum2

 

Adding Polynomials Side by Side
  • How to Add Polynomials Side by Side
    • Step 1: Remove the parentheses.
    • Step 2: Collect like terms.
      • When you collect like terms in a polynomial, you group terms by degree.
    • Step 3: Simplify.

Alg1B 10.2 Add Like Terms

  • Example, using the steps (above)

Alg1B 10.2 AddingSbS

  • Another example (using negative terms)
    • Remember, minus signs are just negatives!
    • Negatives belong to the terms that follow them.

Alg1B 10.2 AddingSbSnegatives

 

Adding Polynomials Vertically
  • How to Add Polynomials Vertically
    • Step 1: Arrange the problem vertically.
    • Step 2: Fill in any missing terms using zero coefficients.
    • Step 3: Add the polynomials.
    • Step 4: Simplify the sum.

Alg1B 10.2 Add Vert1

  • Example, using steps (above)

Alg1B 10.2 Add Vert Zeros

 

Subtracting Polynomials Using Tiles
  • How to Use Tiles to Subtract Polynomials
    • Step 1: Use tiles to show each polynomial.
    • Step 2: Remove subtracted tiles.
    • Step 3: Write a polynomial for the tiles left over.
  • Example, using steps (above)

Alg1B 10.2 Sub Tiles

 

Subtracting Polynomials Side by Side
  • How to Subtract Polynomials Side by Side
    • Step 1: Use the distributive property to change the subtracted polynomial to an added polynomial.
      • Multiply a negative 1 through the 2nd polynomial
      • This changes the sign of each term in the 2nd polynomial
    • Step 2: Solve it like an addition problem.
  • Example, using steps (above)

Alg1B 10.2 SubSbS

 

Subtracting Polynomials Vertically
  • How to Subtract Polynomials Vertically
    • Step 1: Make sure the polynomials are written in descending order.
    • Step 2: Write the polynomials vertically with the like terms aligned.
    • Step 3: Replace any missing terms with terms that have a zero coefficient.
    • Step 4: Multiply the bottom polynomials by (-1).
    • Step 5: Add the polynomials.
  • Example, using steps (above).  Video has no sound.

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