1. Polynomials and Factoring

2. What is a polynomial? Any finite sum of terms. 6x² + 2x – 1 5x – 315
→ trinomial of degree 2
→ binomial of degree 1
→ polynomial of degree 6
→ monomial of degree 4
→ monomial of degree 1

3. Adding & Subtracting
To add or subtract two polynomials, add or subtract the like terms.
Like terms
Any two terms with the same variables and exponents

4. Adding & Subtracting Which of the following are like terms?
1) 7xy and 6x
2) 3y and -4y
3) 5x³ and 3x²
4) 4x²y³ and -2x³y²
→ not like terms (variables are different)
→ These are like terms
→ not like terms (exponents are different)
→ not like terms (exponents)

5. Adding & Subtracting = 4x³ + 9x² - 5x + 6 Find the sum:
(5x³ - x + 2x² + 7) + (3x² + 7 – 4x) + (4x² x³)
Starting with the highest degree, what are like terms?
= 4x³
+ 9x²
- 5x
+ 6

6. Adding & Subtracting = 3x² + 2x + 1 Find the sum:
(2x² + x - 5) + (x + x² + 6)
= 3x²
+ 2x
+ 1

7. Adding & Subtracting Add the following polynomials:
(12x³ + 10) + (18x³ - 3x² + 6)
(4x² - 7x + 2) + (-x² + x – 2)
(8y³ + 4y² + 3y – 7) + (2y² - 6y + 4)
(x³ - 6x) + (2x³ + 9) + (4x² + x³)

8. (12x³ + 10) + (18x³ - 3x² + 6)
= 30x³
- 3x²
+ 16

9. (4x² - 7x + 2) + (-x² + x – 2)
= 3x²
- 6x

10. (8y³ + 4y² + 3y – 7) + (2y² - 6y + 4)
= 8y³
+ 6y²
- 3y
- 3

11. (x³ - 6x) + (2x³ + 9) + (4x² + x³)

12. Polynomials & Factoring

13. Adding & Subtracting
Find the sum:
(2x³ - 3x + 6x² + 7) + (-3x² x) + (2x² - x³)
Starting with the highest degree, what are like terms?
= x³
+ 5x²
- 2x

14. Subtracting So far, the only operation we have used has been addition.
Most of the mistakes made on MCAS tests are on Subtraction problems

15. Subtraction Find the difference (5x² + 2x – 4) – (3x² - 3x + 6)
You must distribute the negative sign over the
polynomial in the immediately parentheses to the right
5x² + 2x – 4
-3x²
+ 6x -6
= 2x²
+ 8x
- 10

16. Subtracting Find the difference between the polynomials.
(5h² + 4h + 8) – (3h² + h + 3)
→ Re-write the equation after distributing the minus sign
5h² + 4h + 8
+ -3h²
- h -3
= 2h²
+ 3h
+ 5

17. Subtracting (-7z³ + 2z – 7) – (2z³ - z – 3) -7z³ + 2z - 7 + -2z³ + z
+ 3
= -9z³
+ 3z
- 4

18. Perimeter
The perimeter of a polygon is the sum of the lengths of its sides. Write a simplified expression for the perimeter of this polygon.
x + 2
4x – 2
2x + 3
3x + 15
Per. =
2x + 3
+ x + 2
+ 4x – 2
+ 3x + 15
Per. =
10x
+ 18

19. Perimeter
The perimeter of a polygon is the sum of the lengths of its sides. Write a simplified expression for the perimeter of this polygon.
x + 4
x - 3
x – 3
x - 1
Per. =
x - 3
+ x + 4
+ x – 3
+ x - 1
Per. = 4x
- 3

20. Polynomials & Factoring

21. So far we have: Added polynomials (4x³ + 2x² + 3) + (2x³ - x² - 2)
Subtracted polynomials
(3x² + 4x + 8) – (2x² + 3x – 5)
3x² + 4x + 8
6x³
+ x²
+ 1
- 2x²
- 3x
+ 5 x²
+ x
+ 13

22. Multiplying Polynomials
When we multiply two binomials
(x + 2) (2x – 4)
Use the F.O.I.L. method
F -
First
O -
Outer
I -
Inner
L -
Last

23. F.O.I.L (x + 2) (2x – 4) = 2x² - 4x + 4x - 8 = 2x² - 8 F - First
O - Outer
x • - 4
I - Inner
2 • 2x
L - Last
2 • -4

24. F.O.I.L (x + 2) (x – 3) = x² + -3x + 2x + -6 = x² + -1x - 6 x • x = x²
2 • -3
= -6

25. Multiply the following binomials.
(2x – 3) (x + 4)
(3x + 1) (x + 5)
(x – 4) (x + 7)
(4x + 4) (2x – 3)

26. F.O.I.L (2x - 3) (x + 4) = 2x² + 8x + -3x + -12 = 2x² + 5x - 12 2x • x
-3 • 4
= -12

27. F.O.I.L (3x + 1) (x + 5) = 3x² + 15x + x + 5 = 3x² + 16x + 5 3x • x
1 • 5
= 5

28. F.O.I.L (x - 4) (x + 7) = x² + 7x + -4x - 28 = x² + 3x - 28 x • x = x²
-4 • 7
= -28

29. F.O.I.L (4x + 4) (2x - 3) = 8x² + -12x + 8x - 12 = 8x² - 4x - 12
4 • -3
= -12

30. Polynomials & Factoring

31. This week: Added polynomials Subtracted polynomials
Multiplied binomials (F.O.I.L.)

32. Adding Polynomials (x² + 3x – 4) + (3x² + 5x – 8) = 4x² + 8x - 12
- 4

33. Subtracting Polynomials
(5x² + 2x – 7) - (2x² + 4x – 3)
= 5x² + 2x - 7
- 2x²
- 4x
+ 3
= 3x²
- 2x
- 4

34. F.O.I.L (2x + 3) (4x - 1) = 8x² + -2x + 12x - 3 = 8x² + 10x - 3
3 • -1
= -3