1. Polynomials
By: Ms. Guarnieri

2. Vocabulary Binomials- TWO TERMS – ONE ADDITION OR SUBTRACTION SIGN
Monomials - a number, a variable, or a product of a number and one or more variables. 4x, 20x2yw3, -3, a2b3, and 3yz are all monomials. ONE TERM – NO ADDITION OR SUBTRACTIONS SIGNS
Binomials- TWO TERMS – ONE ADDITION OR SUBTRACTION SIGN
Trinomials- THREE TERMS – TWO ADDITION OR SUBTRACTION SIGNS
Polynomials – one or more monomials added or subtracted
4x + 6x2, 20xy - 4, and 3a2 - 5a + 4 are all polynomials.

3. Like Terms
Like Terms refers to monomials that have the same variable(s) but may have different coefficients. The variables in the terms must have the same powers.
Which terms are like? a2b, 4ab2, 3ab, -5ab2
4ab2 and -5ab2 are like.
Even though the others have the same variables, the exponents are not the same.
3a2b = 3aab, which is different from 4ab2 = 4abb.

4. Constants are like terms.
Which terms are like? x, -3, 5b, 0
-3 and 0 are like.
Which terms are like? x, 2x2, 4, x
3x and x are like.
Which terms are like? wx, w, 3x, 4xw
2wx and 4xw are like.

5. Adding Polynomials Add: (x2 + 3x + 1) + (4x2 +5)
Step 1: Underline like terms:
(x2 + 3x + 1) + (4x2 +5)
Notice: ‘3x’ doesn’t have a like term.
Step 2: Add the coefficients of like terms, do not change the powers of the variables:
(x2 + 4x2) + 3x + (1 + 5)
5x2 + 3x + 6

6. Adding Polynomials
Some people prefer to add polynomials by stacking them. If you choose to do this, be sure to line up the like terms!
(x2 + 3x + 1)
+ (4x )
(x2 + 3x + 1) + (4x2 +5)
5x2 + 3x + 6
Stack and add these polynomials: (2a2+3ab+4b2) + (7a2+ab+-2b2)
(2a2 + 3ab + 4b2)
+ (7a2 + ab + -2b2)
(2a2+3ab+4b2) + (7a2+ab+-2b2)
9a2 + 4ab + 2b2

7. Adding Polynomials
Add the following polynomials; you may stack them if you prefer:

8. Subtracting Polynomials
Subtract: (3x2 + 2x + 7) - (x2 + x + 4)
Step 1: Change subtraction to addition (Keep-Change-Change.).
(3x2 + 2x + 7) + (- x2 + - x + - 4)
Step 2: Underline OR line up the like terms and add.
(3x2 + 2x + 7)
+ (- x2 + - x + - 4)
2x x + 3

9. Subtracting Polynomials
Subtract the following polynomials by changing to addition (Keep-Change-Change.), then add:

10. Multiplication of Polynomials
Multiplying two polynomials requires the distributive property. The rule for multiplication can be interpreted geometrically.

11. Rectangle Method
(x + 1) (3x + 2) +2 3x
3x2 2x x 3x 2 +1

12. Review: When multiplying variables,
add the exponents!
1) Simplify: 5(7n - 2)
Use the distributive property.
5 • 7n
35n - 10
- 5 • 2

13. 12t4 5) Simplify: - 4m3(-3m - 6n + 4p) 12m4
4) Simplify: 4t2(3t2 + 2t - 5)
12t4 5) Simplify: - 4m3(-3m - 6n + 4p) 12m4
+ 8t3
- 20t2
+ 24m3n
- 16m3p

14. Simplify 4y(3y2 – 1)
7y2 – 1
12y2 – 1
12y3 – 1
12y3 – 4y

15. Simplify -3x2y3(y2 – x2 + 2xy) -3x2y5 + 3x4y3 – 6x3y4

16. Multiplication Rule
FOIL Mnemonic – a memory aid for multiplying two binomials.
Outer
First
Inner
Last

17. You Try It!! 1. 2x2(3xy + 7x – 2y) 2. (x + 4)(x – 3)

18. Problem One 2x2(3xy + 7x – 2y) 2x2(3xy + 7x – 2y)
2x2(3xy) + 2x2(7x) + 2x2(–2y)
Problem One
6x3y + 14x2 – 4x2y

19. Problem Two (x + 4)(x – 3) (x + 4)(x – 3) x(x) + x(–3) + 4(x) + 4(–3)

20. Problem Three (2y – 3x)(y – 2) (2y – 3x)(y – 2)
2y(y) + 2y(–2) + (–3x)(y) + (–3x)(–2)
Problem Three
2y2 – 4y – 3xy + 6x