AIM: SWBAT classify polynomials, write them in standard form and state their degree.

DO NOW:List the terms in each expression.

1) 6x2 + 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, x2 – 9)

Trinomial – three terms (Ex: x2 + 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 – 8y3 ____________ 4) x2 + 6x + 5 _____________

5) x3 – y3 _____________ 6) -5x2y _____________

7) x4 + x3 + 3x2 + x – 8 _____________ 8) x8 – x5 + x4 + 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: 5x3 – 2x2 + 3x + 7

Write each polynomial in standard form.

9) 14x + 2 – 3x2 + 5x3 ____________________________________

10) 8z2 – 2z + 7 – 9z3 ____________________________________

11) 2y – 7y5 + 3y2 + 2 ____________________________________

12) x3 – 2x2 + 7x5 + 4 ____________________________________

13) 2x + 5x2 - 7 ____________________________________

Like Terms – monomials with the same variables with the same exponents. Ex: 5m & 3m, 2x2 & x2, xy3z & -9xy3z

1

For each question circle the terms that are Like-Terms:

1) x2y, 5xy, 10xy2, 3x2y, -2x2y 2) xyz, 46xy, 12zx, -6xy, 8yz

3) 132, 4x2y, 16zp, 4x3h, -12hx3 4) 7ha, 8ah, 9hz2, 5p3x4, 11ah2

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) 3y + y2 + 5y – 9y2 2) 7x + 2x – 5 + 3 3) -6a + 2a + 7a

4) x2 + 5x – 7 – 12x – 13 5) 5(x4 – 3x3 + 2x2) 6) -3(2y – 7y2 + 8) – 5y

7) 4(x + 4) –3x4 + 9x2 -2x + 5 8) n(n + 5) + 6n 9) x(2x + 3) + 4x2 – x + 1

HOMEWORK – Intro to PolynomialsClassify each polynomial as a monomial, binomial, trinomial, or a polynomial.

1) 3x – 5 ___________ 2) x2 - 3x + 4 __________3) 4y __________ 4) -7a + 9b __________

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5) n5 - 7n4 + 5n3 - n + 2 __________ 6) -8x3y6 __________

Find the degree of each polynomial.

7) 5______ 8) 3x2 + 4y2 ______9) 3x4y2z ______ 10) 7x3y – 2y5 _______11) x5 – 8x3 + x6 _______ 12) -3x2 + 7x ________Circle the terms that are like terms.

13) 2xy, ut2x, 2xt2u, 14xz, -7xut2 14) 3x3p, 12pxy, 3x3z, -4px3, 13px3

15) mnr, m2nr, mn2r, mnr2, mn2r 16) 4pqs, 2p2qs, -11pq2s, 13pq2s

17) 13x, 17, 126zu, 31p, -72, 14xy 18) r2, 2r, 3r3, 5r2, 8r3

Simplify each polynomial and write in standard form.

19) 6x – x2 + 4x – 7x2 20) 2y – 8 + 4y – 2

21) -4x4 – x3 + x4 22) 4(y3 - 2y2 + 3)

23) -3(3x3 – 6x + 4x3) – 2x 24) x(3x – 2) + 2x2 – x + 5

25) 2n(n2 – 5n) + 6n3 26) 2(x - 5) + 3(x2 + 6)

DO NOW – Intro to Polynomials

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 ____________________ _________ _______________

4) -5 + 3x2 __________________________ _________ _______________

5) Describe and correct the error made in simplifying the polynomial.3

-5x2 – 6(3x + 2)= -5x2 – 18x – 12= -23x2 – 12

State the degree and the numerical coefficient of each term.6) -7xy2z _________ _________ 7) 12 ________ ________

8) 24xz3 _________ _________ 9) 17y _________ _________

Simplify (distribute and combine like-terms) each polynomial and write your final answer in standard form.10) 2c2 – c2 + 5c 11) 4q3 – 7q5 + 3q – q3 12) -6(2y3 - 4y2 + 1) + 10y2

AIM: SWBAT add and subtract polynomials.

Adding Polynomials To add polynomials: Distribute the positive sign to each term in

parenthesis. This does not change the sign of each term.

Use the commutative property to rearrange the terms so that like terms are beside each other.

When you are rearranging terms, keep the sign with the term. Combine like terms following the rules for adding integers.

Examples: A) (x2 + 9x – 5) + (-4x2 - 12x + 5) B) (8y – 9) + (-6y + 2) x2 + 9x – 5 - 4x2 - 12x + 5 8y – 9 - 6y + 2 x2 - 4x2 + 9x - 12x – 5 + 5 8y - 6y - 9 + 2

Add the following polynomials.

1) (x + 3) + (8 – 5x) 2) (-4x – 7) + (2x + 9)

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-3x2 - 3x 2y - 7

3) (3x + 10) + (6x – 15) 4) (x2 + 5x) + (9x2 + 4x)

5) (-5x2 + 8x + 12) + (3x2 – 4x – 8) 6) (6x2 - 11x - 17) + (9x2 – 12)

Subtracting PolynomialsThe additive inverse is a numbers opposite. To find the opposite or additive inverse multiply by ______.

Find the opposite of the following:

7 _______ -4 _______

-3a _______ 5x _______

(6a – 5b) __________ (-7b + 3c) __________

(-4x + 3y -6z) ________________ (5y – 4x + 7c) __________

To subtract polynomials: Distribute the negative sign to each term in parenthesis.

This changes the sign of each term to its opposite. Combine like terms following rules for adding integers.

Examples:A) (5y – 8) – (3y – 2) B) (4n2 + 11) - (-10n2 + 7) 5y – 8 - 3y + 2 4n2 + 11 + 10n2 - 7 5y - 3y – 8 + 2 4n2 + 10n2 + 11 – 7

Subtract the following polynomials.

1) (7x – 4) – (x + 3) 2) (-6x + 5) – (8x – 2)

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2y - 6 14n2 + 4

3) (3x2 + 10x) – (6x2 – x) 4) (5x2 + 12x – 17) – (13 - 4x2 + 8x)

5) (9x2 + 5x + 7) – (4x2 - 3x + 8) 6) (-11x2 – 12x + 13) – (4x3 + x2 – 6)

HOMEWORK – FINDING THE SUM or DIFFERENCE

Find each sum OR difference.

1) (2a – 6) + (3a + 8) 2) (9m + 7n) + (-4m + 3n)

3) (2x + 4y – 1) + (-x – 7 – 6y) 4) (3p + 2r) + (12r – 2p + 7)

5) (2k + 3kn) + (-6kn + 4k) 6) (7u2 – 10r) + (-3u2 + 8 – 2r)

7) (3x + 2) – (5x – 1) 8) (4r + 2u) – (-7r – 87)

9) (3c + 7d – 5) – (6d + 4 – 2c) 10) (7x – 3y + 9) – (4y – 8)

11) (7ax + 13by + 5) – (-3ax + 4) 12) (2a3 + 7a2b + b3) – (a3 + 7b3)

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DO NOW - Polynomial Review

Classify the following polynomial.

1) 2x2 - 9x + 24 _______________

Write the polynomial in STANDARD FORM and state the degree.

2) 7x5 + x7 – 4x2 ____________________________

Simplify each polynomial.

3) 9x2 + 7x – 3 + 7x2 - 9x + 2 4) -2y(2y - 5y2 + 3) + 6y2 – 9y + 7

State the coefficient of the following monomials and the degree.

5) -8x2y3 ______ ______ 6) 12m2n4 ______ ______7) 24 ______ ______

ADD the polynomial. SUBTRACT the polynomial.

8) (7y2 + 5y – 8) + (8y2 – y + 5) 9) (9x2 – 5x + 25) - (4x2 + 8x + 15)

AIM: SWBAT add or subtract polynomials.

Add or subtract the following polynomials.

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1) (-2x2 + 4x - 12) + (5x2 - 5x) 2) (3y2 - 9y) - (-5y2 + 7y - 7)

3) (3x2 - 2x + 1) - (4x3 - 5x - 8) 4) (6x3 - 2x2 - 12) + (6x2 + 3x + 8)

5) (x2 – x - 4) - (3x2 - 4x + 5) 6) (x3 - x2 + 3) - (3x2 - 4x + 5)

7) (4x2 + 2x – 3) – (2x2 - 5x – 3) 8) (x2 + 5x – 24) + (-x2 - 4x + 9)

9) (x3 + 9x – 5) – (-4x2 - 12x – 5) 10) -d2 + [9d + (2 – 4d2)]

11) (4x2 + 6x + 3) + (3x2 - 3x - 2) + (-4x2 + 3x - 9)

12) (7x2 + 2x + 7) - (4x2 - 2x + 3) + (-5x2 + 6x + 7)

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13) (3x3 - 5x2 - 9) - (5x3 - 5x - 4) - (5x3 - 4x2 - 9)

14) (2x2 - 9x + 8) - (2x3 - 4x2 – 8x - 2) + (-5x3 - 6x – 10)

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)

3) (5ab + 2ac – 6bc) + (-4ac + 2bc) 4) (6x2 – 3x + 1) + (3x3 + 4x2 – 5x)

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5) (2a2 + 4a – 1) – (a – 6a2 + 2) 6) (6r2x + 5rx2) – (9rx2 – 9r2x)

7) (5n2 + 2n – 9) + (3n2 – 4) 8) (3p2 – p – 1) + (p2 + p – 4)

9) (x + 15y – 9z) – (7x – 8y + z) 10) (4r2 – r + 8) – (r2 + 6r – 1)

11) (4x3 + 5x2 – 2x – 5) – (3x3 – 4x + 2) 12) (2mn + 3a + 7d) + (-5mn + 7a)

AIM: SWBAT find the perimeter of geometric shapes. DO NOW: Find each sum or difference.1) (3x + 10) + (6x – 15) 2) (x2 + 9x) – (5x2 – 3x)

3) (-5x2 + 8x + 12) + (3x2 – 4x + 8) 4) (6x2 - 11x - 17) - (9x2 – 8x - 12)

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)

2) Simplify: 2(3ac + 4bc) – 3(5bc – 15ab) – (2ab + bc -2ca)

3) Find the sum of (8a+3b)and(5a−2b−c )

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)

5) (-x3)(-x8) 6) (-y2)(y3) 7) (3x3y3)(7x2y3) 8) (-5xy3)(8xy)

CLASSWORK: To multiply a polynomial by a monomial use the____________________.

Simplify:

9) 2x(3x + 1) 10) –x(x2 – 4) 11) -8x(x5 + x)

12) 3k2(12 – k5) 13) 4q3(3q + 6) 14) -7a2(5a3 – 9a)

Simplify the following monomials. 15) (5x)3 16) (xyz)5 17) (3rs)2 18) (-3xy)3 19) (c2)9

Complete each statement with a monomial.

20) 8a6 + 64a3 = _____(a3 + 8) 21) 18x2y – 9xy2 = _____(2x – y)

22) 12b2 – 9b2c4 = _____(4 – 3c4) 23) 12x3y6 + 20x5y2 = _____(3y4 + 5x2)HOMEWORK – Multiplying Polynomials

Simplify. Write your answer in standard form.1) x(x + 3) 2) 3x(x + 7) 3) 2c(c – 3d)

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4) x2y (x3 – y) 5) -8mn (14m – 8n + 3) 6) x3z2(z2 – 5)

7) xa4(x2a – 8c) 8) 4x2y (2x3y + 15z - 7) 9) 3x3 (y2 + 2x + z)

10) (5x3)2 11) (-3y4)3 12) (5mn3)2

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:

(x + 5)(x + 2) (x – 3)(x + 3)

Multiply the binomials.

1) (y – 6)(y + 7) 2) (x - 7)(x – 2)

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x + 1

2x + 3

3) (3x – 2)(2x + 4) 4) (x + 4)2

*5) (x + 2)(x2 + 3x + 1) 6) Find the area:

Extra Practice – Multiplying Binomials

Multiply the binomials.1) (x – 1)(x – 5) 2) (x + 6)(x + 4) 3) (x – 9)(x + 9)

4) (x – 8)(x + 8) 5) (x + 12)(x + 2) 6) (x – 7)(x – 5)

7) (x + 11)(x + 5) 8) (x - 6)(x – 6) 9) (2x – 4)(3x + 2)

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10) (x – 5)(x + 11) + (x + 10)(x + 9) 11) (3x + 5)(x + 4) + (x – 1)(x + 14)

HOMEWORK – MULTIPLYING BINOMIALS

Simplify the following expressions by multiplying. Make sure your answer is in standard form.

1) -7x(2x + 9) 2) 3s(5s² + 2s – 1) 3) 2x(2x + 5)

4) (x + 5)(x + 1) 5) (y + 4)(y – 7) 6) (x – 10)(x + 3)

7) (a – 4)(a - 5) 8) (2x + 6)(x – 1) 9) (2x + 2)(6x + 1)

10) (x + 6)2 11) (x – 6)2 12) (x + 3)(x2 + 4x + 2)

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)

4) x2y(x3 – y) 5) -8mn(14m – 8n + 3) 6) x3z2(z2 – 5)

7) xa4(x2a – 8c) 8) 4x2y(2x3y + 15z - 7) 9) 3x3(y2 + 2x + z)

10) (5x + 7)(5x – 7) 11) (2x + 4)(3x + 9) 12) (4x + 1)(4x – 2)

13) (x – 2)2 14) (x – 5) (x + 5) 15) (x - 5) ( x – 7)

16) Find the area AND perimeter. x + 8

x + 3

17) 24x3

8x = 18) 60b8

12b6 = 19) 40y5

4y2 = 20) 30m4

10m=

20

21) 25 x5− 10 x3+ 5x

5 x 22) 18 y3+ 8 y2+4 y

2 y

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.

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___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ 3 5 9 2 7 11 6 7 8 9 5 10 10 8 9 1 4 2 7

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

Example 4: Factor 10x3 – 5x2

Find a GCF and factor each expression.

1) 8x2 + 10x 2) 12y - 16 3) -15d5 + 45d3

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4) c3 + c2 – c 5) 6n2 – 30n + 42 6) 18p3 – 63p2 – 9p

7) 100x9 + 50x6 - 75x5 8) 36rs2 – 108r2s3 9) 36k - 30

10) 18x2 – 50y2 11) a7b – a10 12) 18x5 – 48x4 + 56x3 – 86x

HOMEWORK – FACTORING with the GCF

Find the GCF and factor each expression.1) 9x2 – 21x 2) 15x2 + 20x

3) 12x2 + 28x 4) 15x4 – 24x2

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5) 24x4 – 18x 6) 12x3 + 6x2 - 30

7) 4x4 – 22x2 + 18x 8) 21x5 + 35x3 + 49x2

9) 2c5d4 – 3c4 + 4c3 10) 23y10 – 46y7 + 68y2 + 10y

DO NOW Use Mental Math to find binomial products. Look for patterns.

1) (x + 3)(x + 5)

2) (x – 3)(x – 5)

3) (x + 3)(x – 5)

4) (x – 3)(x + 5)

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The FIRST TERM of the trinomial is the

_______________________________________________________________________

The MIDDLE TERM of the trinomial is the

_______________________________________________________________________

The LAST TERM of the trinomial is the

_______________________________________________________________________

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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?

x2 + 2x – 15 = ( x __ ___ ) ( x __ ___ )

Check:

Factor each expression below. 1) x2 + 6x + 5 2) x2 – 15x + 56

3) x2 – 7x – 8 4) x2 + 5x - 36

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Factors of ___ Sum

Factors of ___ Sum

Factors of ___ Sum

5) x2 – 7x + 18 6) x2 – 7x – 18

HOMEWORK – FACTORING TRINOMIALS

Factor each expression, then check your answer.

1) x2 + 9x + 18 2) x2 – 12x + 35

3) y2 – 13y + 12 4) x2 – 11x + 24

5) x2 + 3x – 10 6) y2 + 3y – 4029

same signs

same signs

Different signs more negatives

Different signs more positives

7) m2 – 7m – 8 8) x2 + 5x – 6

9) n2 + 2n – 63 10) a2 – 7a – 18

AIM: SWBAT factor a trinomial.

DO NOW: Match the polynomial to the correct product.1) (x + 2)(x + 6) 2) (x – 3)(x – 4) 3) (x + 3)(x – 4) 4) (x + 2)(x – 6)

A) x2 - 7x + 12 B) x2 – 4x – 12 C) x2 + 8x + 12 D) x2 – x - 12

Factoring a trinomial. That's SUM PRODUCT!

1) x2 + 8x + 15 You need 2 factors of +15 whose sum is +8. 1, 15 3, 5

( )( )

2) x2 - 8x + 12 You need 2 factors of +12 whose sum is -8. 1, 12 2, 6

( )( ) 3, 4

3) x2 - 12x - 28 You need 2 factors of -28 whose sum is -12. 1, 28 2, 14

( )( ) 4, 7

4) x2 + 3x - 40 You need 2 factors of -40 whose sum is +3. 1, 4030

2, 20 ( )( ) 4, 10 5, 8

Factor each trinomial.1) y2 + 9y + 14 2) x2 - 8x + 15 3) w2 + 5w – 6

4) x2 + 2x - 35 5) x2 – 11x + 30 6) x2 + 12x + 36

7) x2 - 8x - 9 8) x2 + 8x + 12 9) x2 - 9x + 14

10) x2 + 2x – 15 11) x2 + 13x + 36 12) x2 – 6x - 7

13) x2 – 4x – 12 14) x2 – 15x + 56 15) x2 – 18x + 72

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HOMEWORK – FACTORING TRINOMIALS

Factor each of the following trinomials.1) x2 + 12x + 32 2) x2 - 7x - 30 3) x2 + 8x - 9

4) x2 + 14x + 40 5) x2 + 5x - 24 6) x2 - 9x + 20

7) x2 + 16x + 15 8) x2 - 9x + 14 9) x2 - 11x + 24

10) x2 + 7x + 12 11) x2 + 5x - 14 12) x2 - 10x + 21

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Review PolynomialsState the degree for each polynomial.

1) 5y2 + 7xy3 ________ 2) 7x3y + 5x2 + 3 ________ 3) 5x2 + 2x + 1 ________Write each polynomial in STANDARD FORM.

4) 6x + 2x2 5) -7 + 5y5 + 8y7 + 9y 6) 3 + 7x + 2x2

_____________________ _____________________ ____________________Identify each polynomial as a monomial, binomial, trinomial or polynomial.

7) 5m2n ________________ 8) 2x + 1 ________________9) -5 + 2x4 + 3x ________________ 10) 4x4 + x3 + x2 + x ________________Simplify.

11) 3(x2 + 9) - 5x - 7 12) (-3x)(6x + 4) + 2x - 9 13) -5(x2 + 7x – 3) +x2 + 5x - 2

Add or Subtract.

14) (5x2 + 2x + 3) + (3x2 + x + 4) 15) (14x2 + 5x + 9) – (3x2 + 3x + 3)

16) (3x2 – 2x – 3) – (5x2 – 2x + 3) 17) (-7x2 + 5x – 2) + (-2x2 + 4x – 3)

18) (3x2 – 5x – 4) + (4x2 – 6x + 2) 19) (-5x2 – 2x – 4) – (-2x2 – 5x + 7)

20) (-8x2 – 4x) – (10x2 + 2x) 21) (-4x2 + 6x – 7) + (-2x2 – 9x – 3)Multiply or Divide.

22) x2(2x3 – 5) 23) -2x(7x – 9) 24) 3y(y7 – 10y)33

25) (x5y3)4 26) (2x3y7)4 27) (-3x4y)3

28)

18 x4 y5

3xy 29)

27 x3 y7

9x2 y 4

30)

14 x5 + 21x3 − 7 x2

7 x2

31) (m + 6) (m – 5) 32) (x - 3) (x - 10) 33) (x + 8)(x + 10)

34) (x – 6)(x - 20) 35) (x – 7)(x - 2) 36) (x + 10)(x - 10)

37) (2x - 1)(3x + 6) 38) (5x - 10)(4x - 3) 39) (x + 9)2

40) (x - 5)2 41) (x + 3)(x2 + 4x + 2) 42) (x – 5)(x2 - 4x + 9)

Factor each of the following.

43) 6x3 + 2x4 44) 15x2 – 24x7 45) 4x2 + 12x6 + 36x8

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6y + 2

2y3y - 4 3y - 4

2x + 5

2x - 1

46) 6x2y5 + 15x3y4 47) x2 – 10x – 24 48) x2 + 14x + 49

49) x2 – 11x + 30 50) x2 + 5x – 24 51) x2 + 10x + 25

52) x2 – 2x – 15 53) x2 + 4x – 12 54) x2 – 6x + 9

Find the PERIMETER AND AREA of each of the following figures.

55) 56)

57) Find the perimeter and area of a rectangular garden that measures (3x) ft by (2x2 + 5) ft.

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