Polynomial – a monomial or sum of monomials Can now have + or – but still no division by a variable. MonomialBinomialTrinomial 13x 13x – 4 6x 2 – 5x + 4

DEGREE OF A MONOMIAL The sum of the exponents of all the variables (not the coefficients) Example: 1.) 3x 4 y 7 2.) 7xy 5 z 3

The greatest degree of any term in the polynomial Do not add the terms degrees Example: 1.) 2x 6 + 3x 3 2.) 3x 2 y 3 + 5xy 6 DEGREE OF A POLYNOMIAL

ADDING AND SUBTRACTING POLYNOMIALS Steps 1. Distribute the sign *Remember – and - = + 2. Combine like terms 1.) (3x 2 – 4x 3 + 8) – (2x 3 – 7x 2 – 5)

EXAMPLES 2.) (2x 2 + 7x – 5) – (3x 2 – 5x + 4) 3.) (7x 5 + 3x 3 + x 7 ) + (4x 7 + 5x 2 + 4x 3 )

TRY 1. (7x 5 – 4x ) + (x 5 + 5x 3 – 5) 2. (x 6 + 6x 4 + 8) – (5x 6 - 3x 4 – 10) 3. (2x 5 – 3x ) – (9x 5 – 4x 3 )

MULTIPLYING A POLYNOMIALS BY A MONOMIAL 1. Distribute When multiplying you add exponents 2. Simplify Combine like terms

EXAMPLES 1.) 2x(3x – 6x) 2.) 5x 2 (7x + 4y)

FACTORING POLYNOMIALS - Group if necessary - Find GCF - Pull it out - Put the “left over” in parentheses

EXAMPLES 1.) 12a a 2.) 4x 2 + 8x

TRY 1.) 15x x 2.) 5x + 30 y 3.) 21dc – 3d

MULTIPLYING POLYNOMIALS Example: (x + 3)(x + 2) Foil First Outside Inside Last

OR MAGIC BOX Example: (x + 3)(x + 2)

Example: (2x 2 + 4x + 3)(x + 2)

SPECIAL PRODUCTS (a + b) 2 = (a + b)(a + b) Examples: 1. (x + 3) 2 = 2. (x + 4) 2 =

(a - b) 2 = (a - b)(a - b) Examples: 1. (x - 3) 2 = 2. (x - 10) 2 =

(a + b)(a – b) = a 2 - b 2 Examples: 1. (x + 3)(x – 3) = 2. (x + 5)(x – 5) =

TRY 1. (6p – 1) 2 2. (5m – 2n) 2 3. (x – 4)(x + 4) 4. (x + 8)(x – 8)

- Get everything to one side(=0) - List factors of c (Last term) - Find ones that add to b (middle term) To solve: factor and then set equal to zero STEPS

1.) x 2 + 5x – 6 2.) x 2 + 7x + 12 EXAMPLES

1.) x 2 – 12x ) x 2 + 3x – 18 TRY

FACTORING TRINOMIALS OF THE FORM AX 2 + BX + C * If all three terms have a GCF pull that out first - Multiply a and c - Find factors that add to b - Split middle term - Group - Pull out GCF - Put into two bubbles

EXAMPLES 1.) 6x x ) 5x x + 10

3.) 2x 2 + 9x – 5 4.) 24x 2 -22x + 3

- Multiply a and c - Find factors that add to b -divide the factors by a - put in bubbles - denominator is the coefficient with x - numerator is the constant FACTORING TRINOMIALS OF THE FORM AX 2 + BX + C: METHOD 2

6x x x 5 = 30 ^ __ + __ = 17 ( )( ) EXAMPLE 1

5x x + 10 EXAMPLE 2

1.) 24x 2 – 22x ) x 2 - 8x – 65 TRY

SPECIAL PRODUCTS a 2 – b 2 = (a + b)(a – b) Examples 1.) n 2 – 25 2.) 36x 2 – 49y 2

TRY 1.) m 2 – 64 2.) 16y 2 – 81z 2 3.) x 2 – 9 4.) a 2 – 36

USING MULTIPLE METHODS Pull out a GCF of all first if possible Then group and factor

EXAMPLES 1.) 5x x -5x – 15 2.) 7x 2 +6x -14x - 12

TRY 1.) 4y 2 – ) 6x x -24x -120