2.1 Polynomials (M2 11.1) Name: ___________________________

Obj.: I will be able to put polynomials in standard form and identify their degree and type. I will be able to add and subtract polynomials.

Vocabularyo Monomial o GCF o Polynomial o Binomialo Trinomial o o Standard Form of a Polynomialo Degree of a Polynomial o Degree of a Monomial

Notes Polynomials

o A ___________________ is a real number, a variable, or their product. The ______________ of a monomial is the _______ of its ___________________.

o The addition and/or subtraction of monomials are called ______________________. The _________________of a __________________ equals the _______________

_____________ of its ____________. Standard form of a polynomials: highest degree to lowest degree. Can be named based on degree or number of terms.

Degree Name Degree Name # of Terms

Name

0 3 11 4 22 3

Adding/Subtracting Polynomialso Combine __________ ____________ (terms with the _________ _________________ and

_____________________)o Adding/Subtracting Polynomials

__________________ Method – _________ _____ like terms; then ___________ the _____________________.

________________________ Method – ______________ like terms; then add the coefficients.

If _______________________ polynomials, add the __________________ of each ________________ in the polynomial being subtracted. (for either method above)

2.1 Polynomials (M2 11.1) Name: ___________________________

Practice

Find the degree of each polynomial. 1. 2a3b5 2. 7 x8 y2

3. x2 y−8 x y2+5 x+ y5 4. y5+3 y2+8 y6

Write the polynomials in standard form then name the polynomial based on its degree and number of terms.

5. x2+3−8 x 6. 9 z4−2 z−7 z2

Simplify each of the following.7. 8 x2−6 x+6

+ 6 x2+8 x−78. c2+3c+9

+ 2c2−4 c−3

9. (8 x2−3 x+3 )+(6 x3−8 x) 10. (6 x4−4 x−12 )+(5 x+9 x4)

11. 6 x2−5 x+1 – (8 x2+3x−4)

12. 4 b2+7b−6 – (5b2−4)

13. (8 x4+4 x3−5 )−(7 x2−4 x+6) 14. (x4−3 x3 )−(8x3+2 x+10)

2.2 Multiplying Polynomials (M2 11.2-3) Name: ___________________________

Obj.: I will be able to multiply polynomials using distribution, FOIL, and the vertical method.

Notes Multiplying a Polynomial by a Monomial

o ______________________ the ___________________ into the polynomial by _________________________ ____________ ____________ of the polynomial by the monomial.

o When multiplying ________ ___________________ with like bases, _______ the _____________________ and ___________________ the _____________________.

Multiplying Polynomialso Distributive property

____________________ one ____________________ ___________ the __________________ __________ distribute the ________________________ into the ___________________.

o _____________ Works only for multiplying two binomials Shortened form of distribution Multiply in the order ___________, ______________, ______________, ____________

o ________________ Method ___________ ____ the two polynomials Multiply the ___________binomial by ___________ ______________ at a time

from the ________________ polynomial. _________ like terms from the products.

2.2 Multiplying Polynomials (M2 11.2-3) Name: ___________________________

Practice

Multiply

1. 3 x2(3x+2) 2. 11 z5(5−2 z)

3. 4 y4(2 y2−5 y+4 ) 4. 6 g(g2+3 g)

5. (x−7)(x+8) 6. (x+9)(x−5)

7. (2a+3)(4a−5) 8. (3 y−6 ) ( y+3 )

9. (5 x+3)(8 x2+4 x+1) 10. (4b+10)(6b2−7b+2)

11. 5 x2−2x+6 * 4 x−1

12. 9 x2+5x−7 * 3 x+2

2.3 Multiplying Special Cases (M2 11.4) Name: ___________________________

Obj.: I will be able to square binomials and find the product of the sum and difference of two terms.

Notes ____________________ Binomials

o _____________________ a binomial by ______________ results in a special pattern in the ___________________.

o The pattern ________________ ________________ depending upon whether the two terms are ______________ or ______________________.

______________________________________ ______________________________________

The product of sum and differenceo Both binomials have the _____________ _______ _____________; one is _________________,

the other is ________________________.o When multiplying in this situation, the ___________________ term

___________________ out. ______________________________________

2.3 Multiplying Special Cases (M2 11.4) Name: ___________________________

Practice

Multiply

1. ( x+4 )2 2. (w−6 )2

3. (9c−9 )2 4. (7 r+2 )2

5. (x+8)(x−8) 6. (a+4)(a−4)

7. (3v−5 p)(3 v+5 p) 8. (8b−6d )(8b+6 d)

9. A square yellow rug has a purple square in the center. The side length of the purple square is x inches. The width of the yellow band that surrounds the purple square is 10 in. What is the area of the yellow band?

2.4 Simple Factoring (M2 11.5) Name: ___________________________

Obj.: I will be able to factor out the greatest common factor from a polynomial. I will be able to factor quadratic expressions with a lead coefficient of one.

Notes Factoring Polynomials

o Find the _____________________ ___________________ ____________________ first, then factor it out of the polynomial.

Put quadratic expression in ________________________ form (______________________)o Find factors of ___ that add up to ___. (________________________________________)o Write the quadratic expression in factored form: ____________________

________, b is _____________, c is _____________________o __________ m and n will be _________________

a = 1, b is _________________, c is __________________o _________ m and n will be ___________________

Where c is _________________________o m will be negative while n is positive.

If b is ____________________, ____ is the _____________________ number If b is ___________________, ____ is the ____________________ number

With Two Variableso Put expression in ________________________ form (______________________)

Find factors of c that add up to b. (m∗n=c∧m+n=b) Write the quadratic expression in factored form: ________________________

o Follow the ___________ ____________________ above for the signs of m and n.

b cfactors of c, add up to b

m n+ +– ++ –– –

2.4 Simple Factoring (M2 11.5) Name: ___________________________

Practice

Factor out the greatest common factor. 1. 49 x2−14 x 2. 54b4+36b2

Factor each polynomial completely.3. v2+20 v+99 4. x2+15x+36

5. w2+2w−24 6. z2+5 z−36

7. n2−30n+56 8. r2−18 r+80

9. x2−8 x−20 10. p2−13 p−30

11. 17 x y6+51 x3 y4 12. 13 x3 y5+52x2 y

2.5 Grouping (M2 11.6,8) Name: ___________________________

Obj.: I will be able to factor quadratic expressions with a lead coefficient other than one. I will be able to factor a cubic expression by grouping.

Vocabulary o Factoring by grouping o Reverse FOIL o Tic-Tac-Toe Method

Notes

Quadratic with __________o _______________________ _____________

Recall that when multiplying two binomials, you get ________ _______________ for the Inner and Outer products.

If the terms in the quadratic have a ___________, you will need to _______________ it out ____________.

Reverse FOIL uses factors of _______ to split the ______ term in the _________________ expression.

Find _______. Find factors of ac that add up to b (_______________________________) Replace _______ with _______ and _______. _____________ the first _________ terms and the __________ two terms. Factor out the ______________ of ______________ ______________. Use the ____________________________ property to write the factors

as the _____________________ of two binomials.o Tic-Tac-Toe Method

Uses a ________________ to organize factoring a quadratic. Factor out any _______________ of the entire expression ________________. Find _____ and its factors that add up to b (_______________________________) ________________________ the terms in the table:

Find the GCF of each ______________________ and ________. The ________ of the _________ ___________ GCFs is one factor, and the ________

of the ______________ __________________ GCFs is the other. Cubic Expressions

o Some polynomials of a degree greater than ____ can be factored. o Factoring by _______________________ should be used when there are _____ terms in

the polynomial.o Always factor out any GCF from the entire expression first. o This method is _______________________to ________________ ______________.

Group _________________________ pairs of terms when the polynomial is written in _____________________ form.

Factor out a GCF from _____________ ______________. Use the __________________ property.

2.5 Grouping (M2 11.6,8) Name: ___________________________

Practice

Factor each polynomial completely. 1. 8 x2+10x+3 2. 4 x2+8 x+3

3. 5 y2+14 y−3 4. 5 x2+24 x−5

5. 25 r3+35 r2−35 r−49 6. 4w3+6w2−6w−9

7. 28 x4−12 x3+56x2−24 x 8. 12m4−15m3+24m2−30m

9. The area of a rectangular knitted blanket is14 x2−3 x−2. What are the possible dimensions of the blanket? Use factoring.

2.5 Grouping (M2 11.6,8) Name: ___________________________

2.6 Factoring Special Cases (M2 11.7) Name: ___________________________

Obj.: I will be able to factor perfect square trinomials and the difference of two squares.

Vocabularyo Perfect-Square Trinomial o Difference of Two Squares

Notes Factoring a perfect-square trinomial

o A ________________________________ trinomial is the result of squaring a binomial. o For every real number a and b,

___________________________________________ ___________________________________________

o How to recognize a perfect-square trinomial: ____________ and __________________ terms are _______________ _______________ The ______________________ term is _______________ the ________________ of

one factor from the ________________ term and one factor from the _______________ term.

Factoring the difference of two squareso The difference of two squares is the ____________________________ of one perfect

square from ______________________.o For every real number a and b, _______________________________________

Summary of Factoring Polynomialso Factor out the GCFo If the polynomial has two terms or three terms, look for a difference of two

squares, a perfect-square trinomial, or a pair of binomial factors.o If the polynomial has four or more terms, group terms and factor to find

common binomial factors.o As a final check, make sure there are no common factors other than 1.

2.6 Factoring Special Cases (M2 11.7) Name: ___________________________

Practice

Factor each polynomial completely. 1. x2−8 x+16 2. x2+4 x+4

3. 16 x2−56 x+49 4. 9m2+18m+1

5. x2−81 6. n2−25

7. 16v2−100 8. 64b2−81

9. The expression36 x2+84 x+49 represents the area of a square. Find the side length of the square.