1) This document discusses polynomial operations and rules including combining like terms, adding, subtracting, and multiplying polynomials. Key terms defined include degree of a polynomial, standard form, leading coefficient, monomial, binomial, trinomial, and polynomial. 2) Examples are provided for combining like terms, adding, subtracting, and multiplying polynomials using the distributive property. Special cases like FOIL (First, Outer, Inner, Last) are explained for multiplying binomials. 3) Practice problems with answers are given for multiplying polynomials.

1. Combine Like Terms
1) 3x – 6 + 2x – 8
2) 3x – 7 + 12x + 10
3) 10xy + 5y – 6xy – 14y
5x – 14
15x + 3
4xy – 9y
Warm up

2. PolynomialPolynomial
Operations &Operations &
RulesRules

3. VOCABULARYVOCABULARY

4. Degree
The exponent for a
variable

5. Degree of the Polynomial
Highest (largest)
exponent of the
polynomial

6. Standard Form
Terms are placed in
descending order
by the DEGREE

7. Leading Coefficient
Once in standard
form, it’s the 1st
NUMBER in front of the
variable (line leader)

8. # of
Terms
Name by # of
Terms
1 Monomial
2 Binomial
3 Trinomial
4+ Polynomial

9. Degree
(largest exponent)
Name by
degree
0 Constant
1 Linear
2 Quadratic
3 Cubic

10. 2 9y− +
Special Names:
Linear
Binomial
Degree Name:
# of Terms Name:
Leading Coefficient:
-2

11. 3
34x
Special Names:
Cubic
Monomial
Degree Name:
# of Terms Name:

12. 2
4 6x x+
Special Names:
Quadratic
Binomial
Degree Name:
# of Terms Name:
Leading Coefficient: 4

13. 3 2
7 2y y y+ −
Special Names:
Cubic
Trinomial
Degree Name:
# of Terms Name:
Leading Coefficient: 1

14. Adding
Polynomials

15. ( ) ( )2 2
2 4 3 5 1x x x x− + + + −
1.
3x2
+ x + 2

16. ( ) ( )2
6 2 8x x+ + −
2.
x2
+ 2x – 2

17. Subtracting
Polynomials

18. When SUBTRACTING polynomials
Drop the 1st
parenthesis then
distribute the NEGATIVE to the 2nd
parenthesis.

19. ( ) ( )2 2
3 10 8a a a a+ − −
3a2
+ 10a – 8a2
+ a
– 5a2
+ 11a
3.

20. ( ) ( )2 2
3 2 4 2 1x x x x+ − − + −
3x2
+ 2x – 4 – 2x2
– x + 1
x2
+ x – 3
4.

21. PRACTICE

22. ANSWERS

23. Multiplying
Polynomials

24. The Distributive Property
Look at the following expression:
3(x + 7) This expression is the sum of x and 7 multiplied by 3.
To simplify this expression we can distribute the multiplication by 3 to each
number in the sum.
(3 • x) + (3 • 7)
3x + 21

25. Multiply: (x + 2)(x – 5)
Though the format does not change, we must still distribute each term of one
polynomial to each term of the other polynomial.
Each term in (x+2) is distributed to each
term in (x – 5).

26. (x + 2)(x – 5)
This pattern for multiplying polynomials is called FOIL.
Multiply the First terms.
Multiply the Outside terms.
Multiply the Inside terms.
Multiply the Last terms.
F
O
I
L After you multiply, collect like terms.

27. Example: (x – 6)(2x + 1)
x(2x) + x(1) – (6)2x – 6(1)
2x2
+ x – 12x – 6
2x2
– 11x – 6

28. -2x(x2
– 4x + 2)
3 2
2 8 4x x x− + −
5.

29. (x + 3) (x – 3)6.

30. (3x – 1)(2x – 4)
2
6 14 4− +x x
7.

31. 8. Find the area of the rectangle.
2
28 96 80+ +x x
7 10+x
4 8+x

32. 9. Find the volume.
3 2
9 18+ +x x x
3+x
6+x
x