Polynomials

Multiplying Monomials  Monomial-a number, a variable, or the product of a number and one or more variables.(Cannot have negative exponent) › Example: Not Example: › -5, ½, a › 3a, a/2 (½ a) 2a/b › A 2 b 3 a + b - 6  Constants-monomials that are real numbers › A number by itself, without a variable (Ex: 4)

When looking at the expression 10 3, 10 is called the base and 3 is called the exponent or power means = 1000

An algebraic expression contains: 1) one or more numbers or variables, and 2) one or more arithmetic operations. Examples: x n

In expressions, there are many different ways to write multiplication. 1)ab 2)a b 3)a(b) or (a)b 4)(a)(b) 5)a x b We are not going to use the multiplication symbol any more. Why?

Division, on the other hand, is written as: 1) 2) x ÷ 3

Multiplying Monomials  To MULTIPLY powers that have the SAME BASE, just simply ADD the exponents and leave the base the same.  Example:  2 3 * 2 5 = 2 8  x 5 * x = x 6 (x is the same as x 1 and = 6)

Simplify 1. (-7c 3 d 4 ) (4cd 3 ) = -28c 4 d 7 2.(5a 2 b 3 c 4 ) (6a 3 b 4 c 2 ) = 30a 5 b 7 c 6

Find the Power of a Power  To find the power of a power, multiply the exponents.  Example:  (2 2 ) 3 = 2 6

Simplify 1.(p 3 ) 5 = p 15 2.[(3 2 ) 4 ] 2 = 3 16

Power of a Product -To find the power of a product, find the power of each factor and multiply. (a b ) m = a m b m EXAMPLE: (-2xy) 3 = (-2) 3 x 3 y 3 = -8x 3 y 3 SIMPLIFY the following: 1). (4ab) 2 2). (3y 5 z) 2 3). [(5cd 3 ) 2 ] 3 4). (x + x) 2 5). (x 3 ∙x 4 ) 3

SIMPLIFYING MONOMIAL EXPRESSIONS  To simplify an expression involving monomials, write an expression in which:  1. Each base appears exactly once.  2. There are no powers of powers.  3. All fractions are in simplest form.  SIMPLIFY ( ⅓ xy 4 ) 2 [(-6y)²]³ →(Remember: Start within your  parentheses and work your way  out)

Dividing Monomials

Dividing Powers with the Same Base  To DIVIDE powers that have the SAME BASE, SUBTRACT the exponents.  Quotient of powers: For all integers m and n and any nonzero number a, a m = a m-n.  a n  Example: Simplify a ⁴ b⁷ = a 4-1 b 7-2 = a³ b⁵  a b²   

Power of a Quotient  - For any integer m and any real numbers a and b,  b ≠ 0, ( a / b ) m = a m / b m.   Simplify [ 2a³b ⁵ ] 3 = (2a³b⁵)³  3b 2 ( 3b²)³  = 8 a 9 b 15  27 b 6  = 8 a 9 b 9  27

Power of Zero and Negative Exponents Zero Exponent : For any nonzero number a, a 0 = 1.  Example: 3 0 = 1, x 0 = 1 Negative Exponent Property : For any nonzero number a and any integer n, a - n = 1 and 1 = a n. a n a -n  Example: 4 -2 =  Example: 1 =

→The simplified form of an expression containing negative exponents must contain only positive exponents / 3 19 Answer: 3 -6 = 1 / (y³z 9 ) / (yz²) Answer: y 2 z 7 3. (30h -2 k 14 ) / (5hk -3 ) Answer: 6k 17 h 3

 1. b (-x -1 y) 0  b -5 4w -1 y 2  3. (6a -1 b) 2 4. s -3 t -5  (b 2 ) 4 (s 2 t 3 ) -1  5. (2a -2 b) -3  5a 2 b 4

Stacey has to pick an outfit. She has 6 dresses, 12 necklaces, and 10 pairs of earrings. How many different outfits can she choose from if she wears 1 dress, 3 necklaces, and a pair of earrings?

Polynomials

 A polynomial is a monomial or a sum of monomials.  Types of polynomials  Binomial: sum or difference of two monomials  Trinomial: sum or difference of three monomials.

Degrees  Degree of a monomial-the sum of the exponents  Example: the degree of 8y 4 is 4, the degree of 2xy 2 z 3 is 6 (because if you add all the exponents of the variables you get 6)

Degrees  Degree of a polynomial-the greatest degree of any term in the polynomial  Find the degree of each term, the highest is the degree of the polynomial  Example: 4x 2 y 2 + 3x  Find the degree of each term  4x 2 y 2 has a degree 4  3x 2 has a degree of 2  5 has no degree  The greatest is 4, so that’s the degree of the polynomial.

Arrange Polynomials  Arrange Polynomials in ascending or descending order  Ascending-least to greatest  Descending-greatest to least  Example: 6x 3 –12 + 5x in descending order.  6x 3 + 5x –12

Adding and Subtracting Polynomials

 When adding or subtracting polynomials remember to combine LIKE TERMS.  Example:  (3x 2 – 4x + 8) + (2x – 7x 2 – 5)  Notice which terms are alike…combine these terms. (They have been color coded)  3x 2 – 7x 2 = -4x 2  – 4x + 2x = -2x  8 – 5 = 3  So the answer is… -4x 2 - 2x + 3  Be sure to put the powers in descending order.

1. Add the following polynomials: (9y - 7x + 15a) + (-3y + 8x - 8a) Group your like terms. 9y - 3y - 7x + 8x + 15a - 8a 6y + x + 7a

2. Add the following polynomials: (3a 2 + 3ab - b 2 ) + (4ab + 6b 2 ) Combine your like terms. 3a 2 + 3ab + 4ab - b 2 + 6b 2 3a 2 + 7ab + 5b 2

3. Add the following polynomials using column form: (4x 2 - 2xy + 3y 2 ) + (-3x 2 - xy + 2y 2 ) Line up your like terms. 4x 2 - 2xy + 3y 2 +-3x 2 - xy + 2y 2 _________________________ x 2 - 3xy + 5y 2

4. Subtract the following polynomials: (9y - 7x + 15a) - (-3y + 8x - 8a) Rewrite subtraction as adding the opposite. (9y - 7x + 15a) + ( + 3y - 8x + 8a) Group the like terms. 9y + 3y - 7x - 8x + 15a + 8a 12y - 15x + 23a

5. Subtract the following polynomials: (7a - 10b) - (3a + 4b) Rewrite subtraction as adding the opposite. (7a - 10b) + ( - 3a - 4b) Group the like terms. 7a - 3a - 10b - 4b 4a - 14b

6. Subtract the following polynomials using column form: (4x 2 - 2xy + 3y 2 ) - (-3x 2 - xy + 2y 2 ) Line up your like terms and add the opposite. 4x 2 - 2xy + 3y 2 + ( + 3x 2 + xy - 2y 2 ) x 2 - xy + y 2

Add or Subtract Polynomials 1.(5y 2 – 3y + 8) + (4y 2 – 9) Answer : 9y 2 –3y –1 2.(3ax 2 – 5x – 3a) – (6a – 8a 2 x + 4x) Answer : 3ax 2 – 9x – 9a + 8a 2 x

Find the sum or difference. (5a – 3b) + (2a + 6b) 1.3a – 9b 2.3a + 3b 3.7a + 3b 4.7a – 3b

Find the sum or difference. (5a – 3b) – (2a + 6b) 1.3a – 9b 2.3a + 3b 3.7a + 3b 4.7a – 9b

Multiplying Polynomials

Multiplying a Polynomial by a Monomial

Examples 1. -2x 2 (3x 2 – 7x + 10)  Notice the –2x 2 on the outside of the parenthesis……you must distribute this.  -2x 2 * 3x 2 = -6x 4  -2x 2 * -7x = 14x 3  -2x 2 * 10 = -20x 2  Answer : -6x x 3 – 20x 2

Examples 2. 4(3d 2 + 5d) – d(d 2 –7d + 12)  Notice you have to distribute the 4 and –d  4 * 3d 2 = 12d 2  4 * 5d = 20d  -d * d 2 = -d 3  -d * -7d = 7d 2  -d * 12 = -12d  Put it all together….  12d d –d 3 + 7d 2 – 12d  Notice the like terms….  Answer : -d d 2 + 8d

Multiplying Two Binomials  Example:  (x + 3) (x + 2)  This can be done a number of ways.  Use either FOIL or Box Method

FOIL  (x + 3) (x + 2)  F-Multiply the First terms in each  x * x = x 2  O-Multiply the Outer terms  x * 2 = 2x  I-Multiply the Inner terms  3 * x = 3x  L-Multiply the Last terms  3 * 2 = 6  Answer: x 2 + 5x + 6

Box Method

Combine  Add the two that are circled  Answer:  x 2 + 5x + 6

Polynomials (4x + 9) (2x 2 – 5x + 3)  Multiply 4x by (2x 2 –5x + 3)  4x * 2x 2 = 8x 3  4x * -5x = -20x 2  4x * 3 = 12x  Multiply 9 by (2x 2 –5x + 3)  9 * 2x 2 = 18x 2  9 * -5x = -45x  9 * 3 = 27

Put it all Together  8x 3 – 20x x + 18x 2 –45x + 27  Now combine like terms  Answer:  8x 3 –2x 2 –33x + 27

Special Products  A. Square of a Sum: The square of a + b is the  square of a plus twice the product of a and b plus  the square of b.  Symbols: (a + b)² = (a + b)(a + b)  = a² + 2ab + b²  Example: (x + 7)² = x² + 2(x)(7) + 7²  = x² + 14x + 49  Find each product:  1). (4y + 5)² 2). (8c + 3d)²

 B. Square of Difference: The square of a – b is the  square of a minus twice the product of a and b  plus the square of b.  Symbols: (a – b)² = (a – b)(a – b)  = a² - 2ab + b²  Example: (x – 4)² = x ² - 2(x)(4) + 4²  = x² - 8x + 16  Find each product:  1). (6p – 1)² 2). (5m³ - 2n)²

 C. Product of a sum and a difference: The product  of a + b and a – b is the square of a minus the  square of b.  Symbols: (a + b)(a – b) = (a – b)(a + b)  = a² - b²  Example: (x + 9)(x – 9) = x² - 9²  = x² - 81  Find each product:  1). (3n + 2)(3n – 2) 2). (11v – 8w²)(11v + 8w²)

Summary:  Square of a Sum………(a + b)² = a² + 2ab +b²  Square of a Difference…(a – b)² = a² - 2ab +b²  Product of a Sum and a Difference …………….(a – b)(a + b) = a² - b²

Guided Practice:  1. (a + 6)² 2. (4n – 3)(4n – 3)  3. (8x – 5)(8x + 5) 4. (3a + 7b)(3a – 7b)  5. (x² - 6y)² 6. (9 – p)² 7. (p + 3)(p – 4)(p – 3)(p + 4)

Examples 3. y(y – 12) + y(y + 2) + 25 = 2y(y + 5) – 15  Distribute y, y and 2y  y * y = y 2  y * -12 = -12y  y * y = y 2  y * 2 = 2y  Don’t forget the +25  2y * y = 2y 2  2y * 5 = 10y  Don’t forget the -15

 Now you have…….  y 2 – 12y + y 2 + 2y + 25 = 2y y –15  Combine like terms….  2y 2 –10y + 25 = 2y y – 15  Now you have to solve because you have an equals sign  Answer : y = 2