1. Operating Polynomials
Section 5.3
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5.3 - Operating Polynomials

2. Add/Subtract Polynomials
Keep It
Simple,
Students
Combine like terms when adding or subtracting polynomials
Create a table/chart when simplifying polynomials (for the visual learners)
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5.3 - Operating Polynomials

3. 5.3 - Operating Polynomials
Example 1
Add (x2 – 5x + 6) + (3x2 + 4x – 9)
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5.3 - Operating Polynomials

4. 5.3 - Operating Polynomials
Your Turn
Add (3x3 – y + 2x – 5) + (x + y + 5)
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5.3 - Operating Polynomials

5. 5.3 - Operating Polynomials
Example 2
Subtract (x2 – 5x + 6) – (3x2 + 4x – 9)
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6. 5.3 - Operating Polynomials
Your Turn
Subtract (7x2 + 4x + 7) – (–8x + 2)
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5.3 - Operating Polynomials

7. 5.3 - Operating Polynomials
Example 3
Multiply 5x(2x3 + 6)
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8. 5.3 - Operating Polynomials
Your Turn
Multiply –5x3(2x2 – 9x + 2)
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5.3 - Operating Polynomials

9. 5.3 - Operating Polynomials
Multiplying Factors
FOIL Method:
Multiply the first term in the first bracket and first term in the second bracket (FIRST)
Multiply the first term in the first bracket and second term in the second bracket (OUTER)
Multiply the second term in the first bracket and first term in the second bracket (INNER)
Multiply the second term in the first bracket and second term in the second bracket (LAST)
Simplify the expression
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5.3 - Operating Polynomials

10. 5.3 - Operating Polynomials
Example 4 (using FOIL)
Multiply (x + 2)(x + 3)
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5.3 - Operating Polynomials

11. 5.3 - Operating Polynomials
Example 4 (using Box)
Multiply (x + 2)(x + 3)
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12. 5.3 - Operating Polynomials
Example 5
Multiply (4x + 3)(3x – 5)
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13. 5.3 - Operating Polynomials
Your Turn
Multiply (x4 + 2)(3x2 – 1)
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14. 5.3 - Operating Polynomials
Review
Add (x – 3) + (–x2 + 2x + 4)
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5.3 - Operating Polynomials

15. Example 6 (using Distribution)
Multiply (x – 3)(–x2 + 2x + 4)
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16. 5.3 - Operating Polynomials
Example 6 (using Box)
Multiply (x – 3)(–x2 + 2x + 4)
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17. 5.3 - Operating Polynomials
Example 7
Multiply (x + 2)(3x2 – 4x + 1)
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18. 5.3 - Operating Polynomials
Your Turn
Multiply (2x3 – 3x + 4)(x2 + 1)
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19. 5.3 - Operating Polynomials
Example 8
Multiply (x – 5)(x + 1)(x + 3)
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20. 5.3 - Operating Polynomials
Example 8
Multiply (x – 5)(x + 1)(x + 3)
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21. 5.3 - Operating Polynomials
Your Turn
Multiply (2x + 1)(3x – 2)(4x – 3)
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5.3 - Operating Polynomials

22. 5.3 - Operating Polynomials
Pascal’s Triangle
Named after Blaise Pascal, a famous French Mathematician and Philosopher
Taken from the Fibonnacci’s sequence
Viewed through the process of factoring variable binomials
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5.3 - Operating Polynomials

23. Steps in Binomial Expansion
Plug the equation using Binomial Theorem, using Pascal’s Triangle to determine the coefficients
Follow the countdown method for the parenthesis
The coefficients follow Pascal's Triangle
The first given number or variable increase left to right
The second number or variable decreases left to right
Simplify
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5.3 - Operating Polynomials

24. 5.3 - Operating Polynomials
Pascal’s Triangle
(x + y)0 = 1
Row 0
(x + y)0 1
(x + y)1 = x + y
Row 1
(x + y)1 1 1
(x + y)2 =
x2 + 2xy + y2 1 2 1
Row 2
(x + y)2
(x + y)3 =
x3 + 3x2y + 3xy2 + y3 1 3 3 1
Row 3
(x + y)3
(x + y)4 =
x4 + 4x3y + 6x2y2 + 4xy3 + y4 1 4 6 4 1
Row 4
(x + y)4
(x + y)5 =
x5 + 5x4y + 10x3y2 + 10x2y3 + 5xy4 + y5 1 5 5 1 10 10
Row 5
(x + y)5
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25. 5.3 - Operating Polynomials
Example 9
Multiply (x + y)4 1 2 3 4 6 5 10
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26. 5.3 - Operating Polynomials
Example 9 1 1 1
Multiply (x + y)4 1 2 1 1 3 3 1 1 4 6 4 1 1 5 5 1 10 10 4 3 2 2 3 1 4 x + 4 x y 6 x y 4 1 + + x y + y
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5.3 - Operating Polynomials

27. 5.3 - Operating Polynomials
Example 10
Multiply (x – 5)3
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28. 5.3 - Operating Polynomials
Example 11
Multiply (2x – 1)3
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29. 5.3 - Operating Polynomials
Your Turn
Multiply (3x – 4)3
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30. 5.3 - Operating Polynomials
Example 12
Mr. Silva manages a manufacturing plant. From 1990 through 2005 the number of units produced (in thousands) can be modeled by N(x) = 2x2 + 2x + 3. The average cost per unit (in dollars) can be modeled by C(x) = –4x2 –.1x + 3. Write a polynomial T(x) that can be used to model the total costs.
Total cost is the product of the number of units and the cost per unit.
T(x) = N(x)  C(x)
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31. 5.3 - Operating Polynomials
Your Turn
The number of items is modeled by 0.3x x + 2, and the cost per item is modeled by g(x) = –0.1x2 – 0.3x Write a polynomial c(x) that can be used to model the total cost. Total cost is the product of the number of units and the cost per unit.
y = –0.03x4 – 0.1x x2 – 0.1x + 10
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5.3 - Operating Polynomials

32. 5.3 - Operating Polynomials
Assignment
Worksheet
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5.3 - Operating Polynomials