AIM: SWBAT classify polynomials, write them in standard form and state their degree. DO NOW : List the terms in each expression. 1) 6x 2 + 3x – 9 _____________________ 2) 3x – 5 ___________________ 3) 15x + 16y – 8 + 12x – 5y_____________________ 4) 5xy – y ___________________ CLASSWORK : Polynomials – a monomial or the sum/difference of monomials. Each monomial in a polynomial is called a term. Types of Polynomials: Monomials – one term (Ex: -2x, 4) Binomial – two terms (Ex: 3x + 5, x 2 – 9) Trinomial – three terms (Ex: x 2 + 5x + 4) If a polynomial has more than three terms, it is simply called a polynomial. Classify the polynomial as a monomial, a binomial, a trinomial, or a polynomial. 1) 5m _____________ 2) 2x + 1 _____________ 3) 4 + 3y – 8y 3 ____________ 4) x 2 + 6x + 5 _____________ 5) x 3 – y 3 _____________ 6) -5x 2 y _____________ 7) x 4 + x 3 + 3x 2 + x – 8 _____________ 8) x 8 – x 5 + x 4 + 7 _____________ Standard Form – a polynomial in one variable with no like terms, and having exponents of the variables arranged in descending order. Constant terms are always last in standard form. Ex: 5x 3 – 2x 2 + 3x + 7 Write each polynomial in standard form. 9) 14x + 2 – 3x 2 + 5x 3 ____________________________________ 10) 8z 2 – 2z + 7 – 9z 3 ____________________________________ 1
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AIM: SWBAT classify polynomials, write them in standard form and state their degree.
Standard Form – a polynomial in one variable with no like terms, and having exponents of the variables arranged in descending order. Constant terms are always last in standard form.
To find the DEGREE of a monomial you add the exponents of its variables. A constant has a degree of 0.
Ex. -6r2 has a degree of 2; 12 bc8 has a degree of 9. The exponent of b is 1. You add 1 +
8 = 9Find the degree of each monomial.1) 11m ______ 2) y2 ______ 3) 12 ______ 4) 3m2n ______ 5) -2x2yz4_______
To find the degree of a polynomial you find the degree of each term and choose the largest. EX. 4 + 3a – 8a3 The degree of the polynomial is 3. The last term, -8a3, had the largest sum of exponents.
Find the degree of each polynomial.
6) -5x3yz2 _______ 7) x3 – y3 _______ 8) 5ab2 – 2a + b2 _______ 9) x + 5 ____Simplify each polynomial write your final answer in standard form.
1) Like terms are terms that contain the __________ variables with the ________ exponents. Give an example of like terms. __________________________
Write each polynomial in standard form, state the degree, and classify it.2) 4 + b2 – 8b _________________________ _________ _______________3) 11 + 2n4 – 7n + 5n2 ____________________ _________ _______________
15) Camilla is putting ribbon around the edge of her scrapbook. The dimensions of the rectangular page can be represented by 15c + 3 inches wide by 21c + 1 inches long. How much ribbon will Camilla need to go once around the perimeter of her scrapbook? (Draw a diagram)
HOMEWORK – ADD or SUBTRACT POLYNOMIALS
Add or subtract the following polynomials.1) (x2 + 3x + 2) + (3x2 + 4x – 9) 2) (6m2 + 2m – 3) – (7m2 + 4)
CLASSWORK:Perimeter is the distance around the outside of a polygon. You find perimeter by adding up all the sides of a polygon.
Find the perimeter of each polygon. Show all work step-by–step.1) 2)
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3) 4)
5) 6)
7) Jeremiah is building rectangular dog run. He has determined the length will be 8x + 12 feet and the width will be 3x + 1 feet. Write an expression that Jeremiah can use to calculate how much fencing he needs for the perimeter of the dog run. (Draw a diagram)
8) Tanya is making a piece a modern art. She wants to paint a violet stripe around the edge of the square canvas. The edge of the canvas can be represented by 9a + 2. What is the perimeter of Tanya’s canvas? (Draw a diagram)
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9 ) Jorge has a triangular fire pit in his backyard. In order to keep his dog safe he wants to install a small fence around the fire pit. The longest side of the fire pit is 3 feet long. The two remaining sides can be represented together by the expression 7x – 2. How much fencing does Jorge need to purchase to go around the entire perimeter of the fire pit?
HOMEWORK - PERIMETER
1) Simplify the following expression: (8x−7 )−(6−2x )+(4x+11)
4) Find the perimeter of a rectangle if the length is (x2−3x+2 ) and the width is (3x−7 )
5) Find the perimeter of an equilateral triangle if each side is 5x3 + 3y.
6) Find the perimeter of an isosceles triangle if the base measures 7xy + 9x and each of the other sides measures 2xy – 5x.
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7) Find the perimeter of a square that has a side length of 3x2 +7x.
8) SUBTRACT (3a+4b−2c ) FROM (13a + b – c)
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AIM: SWBAT multiply polynomials.
DO NOW: Review the Laws of Exponents:1) When you multiply powers of the same base, you _____________ the exponents.2) When you raise a power to another power, you _____________ the exponents. 3) When you divide powers of the same base, you _____________ the exponents.
Multiply the following monomials.1) (-4x)(5x3) 2) (-16x)(-3x9) 3) (-x2)(-3x) 4) (3x)(-2x3)
13) Fill in the blank with the correct number.A. (x3y2)2 = x___y4 B. (2m4)___ = 16x16 C. 3x___ (x3 - 5) = 3x6 - 15x___
14) George has a rectangular garden measuring (2x) feet by (5x - 2) feet. Find the area AND perimeter of the garden.
Area Perimeter
15) Peter found that one side of a greeting card measured 4x and another side measured 2x + 3. Find the area of the greeting card.
AIM: SWBAT use the Distributive Property to multiply polynomials.
DO NOW: Simplify each expression.1) 5(9x – 8) 2) 2x(-9x2 - 2x) 3) 4y2(3y3 – 4y + 5)
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4) 6a2b3(5ab – 4b5) 5) 4x2 – 2(x2 – 8x + 9) – 8
Multiplying BinomialsTo multiply a monomial by a polynomial, we use the ______________________Property. -This property can also be used to multiply binomials.-Remember, an expression is fully simplified when there are no parentheses and no like terms. Example 1: (x + 3)(x + 2) Example 2: (x + 4)(x – 5)
What acronym can be used to help us remember how to multiply binomials?
Another way to multiply binomials is by using a chart:
13) Describe and correct the error made in finding the product of (2x -3) and (x + 7)
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x 72x 2x2 14x3 3x 21
Answer: (2x – 3) (x + 7) = 2x2 + 17x + 21
AIM: SWBAT practice multiplying and dividing monomials and polynomials. DO NOW: Simplify each of the following. 1) (2x2)(3x5) 2) (2x2) (x2 + 5x + 7) 3) (2x2)4 4) (x + 4) (x – 3)
5) x6
x2
6) y7
y6 7) n6
n5 8) y9
y4
9) b8 c5
b3 c2 10) a3b7
a3b6 11) 6y6
3y4 12) 12m5n2
4mn
13) 2x5 + 6 x3 + 12x
2x 14) 12n3 + 36n2 + 3n
3n
15) Penny has a rectangular garden that measures x + 4 feet by x - 2 feet. Find the perimeter AND area of the garden.
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CLASSWORK: Simplify each. Write your answer in standard form.1) x(x + 3) 2) 3x(x + 7) 3) 2c(c – 3d)
Extra Practice - Division of Trinomials by Monomials
Divide.
1)xy3 z + 4 y2 z + 6 xyz
yz 2)x3 y − 2 xy + xy 2
xy
3)15a3 + 3a2b − 6ab
3a 4)4 xyz + 12xyz 2 − 6 xy2
2 xy
5)8uv +10uv2w − 8uvw
2uv 6)x4 y − 2 x3 y − x3 y3 z
x2 y
HW: Multiply the binomials in questions 1-11. Match that answer to the correct letter of the alphabet. Enter that letter of the alphabet on the blank corresponding to the problem number.
A B C D E F G H 3x2 + 2x – 1 x2 + 3 2x2 – 12 x2 – 78 x2 + 9x – 36 x2 + 7x – 78 3x2 – 1 x2 – 36 I J K L M N O P x2– 12x + 36 x2 + 36 20x2 – 36 x2 + 4x + 3 x2 + 36 x2 – 16 x2 + 10x + 25 2x2 – 25
Q R S T U V 2x2 + 25 2x2 – 5x – 25 2x2 + 5x – 12 202 – 63x + 36 x2 – 9x – 36 x2 + 25
W X Y Z x2 + 16 2x2 – 5x + 12 3x2 + 1 0
1) ( x + 3 ) ( x + 1 ) 7) ( 4x – 3 ) ( 5x – 12 )
2) ( 2x – 3 ) ( x + 4 ) 8) ( x – 3 ) ( x + 12 )
3) ( x + 13 ) ( x – 6 ) 9) ( 2x + 5 ) ( x – 5 )
4) ( 3x – 1 ) ( x + 1 ) 10) ( x – 4 ) ( x + 4 )
5) ( x – 6 )2 11) ( x + 5 )2
6) ( x – 12 ) ( x + 3 )
Divide:
12) 16b8 c5
4b8c2 13) 20n5m9
20nm7 14) 15r5+ 10r8− 5 r2
5r 15) 27x5 t 21x4−9 x3x
AIM: SWBAT factor an expression by finding a greatest common factor (GCF).
DO NOW: Find the Greatest Common Factor (GCF) of each pair.
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4x 2 + 14x 4
_______ ( + ) GCF remaining factors
1) 18 and 24 2) 12 and 16 3) 144 and 48 4) 15 and 35
5) x and x3 6) x3 and x8 7) 2x and 6x5 8) 24x2 and 36x
Factoring an expression by finding the GCF is reversing the Distributive Property.To factor out a GCF:1) Find the GCF of ALL terms in the expression.2) Divide each term of the expression by the GCF.3) Rewrite the expression as the product of the GCF and the remaining factors.Example 1: Factor 4x2 + 14x4
GCF:_______________ Divide each term by the GCF:
Rewrite as the product of the GCF and the remaining factors:
Example 2: Factor 6x4 – 60x2 Example 3: Factor: 9x6 + 81x3 – 27x
AIM: SWBAT reverse the Distributive Property in order to factor trinomials.In most cases the product of two binomials is a trinomial. The PRODUCT of the FIRST terms in the binomials gives you the FIRST term of the trinomialThe SUM of the PRODUCTS of the OUTER and INNER terms of the binomials gives you the middle term of the trinomial.The PRODUCT of the LAST terms of the binomials gives you the LAST (constant) term in the trinomial.Multiplying binomials with the same operations.(addition)(addition) all terms will be ____________________Ex. (x + 4)(x + 9) = ___________________________________ (x + 6)(x + 1) = ___________________________________(subtraction)(subtraction) only middle term will be ____________________Ex. (x - 4)(x - 9) = ___________________________________ (x - 6)(x - 1) = ___________________________________Multiplying binomials with different operations.(addition)(subtraction)Last (constant) term will always be ____________________. Middle term will be whatever you have more of. Ex. (x - 9)(x + 4) = ___________________________________ (x + 6)(x - 1) = ___________________________________Keep these patterns in mind when you are factoring trinomials. Fill-in the missing information to complete the examples below:1) x2 + 5x + 6 = (x ___ 2) ( x ___ 3) 2) x2 – 7x + 10 = (x ___ 2 ) ( x ___ 5 )3) x2 + 3x - 54 = (x ___ 6 ) (x ___ 9) 4) x2 + 12 x + 20 = (x + ___ ) (x + ___ )5) x2 – 10x + 24 = (x – 4 ) (x - ___ ) 6) x2 – 2x - 48 = (x – 8 ) (x + ___ )7) x2 – 5x - 50 = (x __ 5 ) (x __ 10 ) 8) x2 + 8x - 9 = (x __ ___ ) (x __ ___ )
To find the factors of a trinomial we REVERSE the multiplication process. We look for factors of the constant term that have a SUM of the coefficient of the linear term.
Example 1: What is the factored form of x2 + 8x + 15?
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Step 1—We need to find factors of ______ that have a sum of _____.Step 2—Decide upon the signs of the factors. _______Step 3—Find the factors/Check by Multiplying the Binomials (mental math is ok)
x2 + 8x + 15 = ( x __ ___ ) ( x __ ___ )
Check:
Example 2: What is the factored form of x2 – 11x + 24?
x2 – 11x + 24 = ( x __ ___ ) ( x __ ___ )
Check:
Example 3: What is the factored form of x2 + 2x – 15?