Polynomials and Factoring

Adding and Subtracting Polynomials

Objective: To classify, add, and subtract polynomials.

Objectives• I can find the degree of a monomial.

• I can adding and subtract monomials.

• I can classify polynomials.

• I can add polynomials.

• I can subtract polynomials.

Vocabulary• A monomial is a real number, a variable, or a product of a real

number and one or more variable with whole-number exponents.

• The degree of a monomial is the sum of the exponents of its variables.

• The degree of a nonzero constant is 0. Zero has no degree.

Finding the Degree of a MonomialWhat is the degree of each monomial?

1. 5x2. 6x3y2

3. 44. 8xy5. –7y4z6. 117. 2b8c2

8. –3

PracticeFind the degree of each monomial

1. 3x

2. 20

3. 0

Adding and Subtracting MonomialsWhat is the sum or difference?

1. 3x2 + 5x2

2. 4x3y – x3y3. –6x4 + 11x4

4. 2x2y4 – 7x2y4

5. 3t4 + 11t4

6. 7x2 – 2x2

7. 2m3n3 + 9m3n3

8. 5bc4 – 13bc4

PracticeSimplify:

Vocabulary• A polynomial is a monomial or a sum of monomials.

• Standard form of a polynomial means that the degrees of its monomial terms decrease from left to right.

• The degree of a polynomial in one variable is the same as the degree of the monomial with the greatest exponent.

• A binomial has two terms that are either added or subtracted.

• A trinomial has three terms that are either added or subtracted.

Vocabulary

Classifying PolynomialsWrite each polynomial in standard form. What is the name of the polynomial based on its degree and number of terms?

1. 3x + 4x2

2. 4x – 1 + 5x3 + 7x3. 2x – 3 + 8x2 4. 6x2 – 13x2 – 4x + 45. 5y – 2y2 6. 3z4 – 5z – 2z2

7. c + 8c3 – 3c7

8. x2 + 4 – 3x

PracticeWrite each in standard form and name the polynomial:

Adding PolynomialsA researcher studied the number of overnight stays in U.S. National Park Service campgrounds and in the backcountry of the national park system over a 5 year period.

Campgrounds: –7.1x2 – 180x + 5800 Backcountry: 21x2 – 140x + 1900

In each polynomial, x = 0 corresponds to the first year in the 5 year period. What polynomial models the total number of overnight stays in both campgrounds and backcountry?

Adding PolynomialsA nutritionist studied the U.S. consumption of carrots and celery and of broccoli over a 6 year period. The nutritionist modeled the results, in millions of pounds, with the following polynomials.

Carrots & Celery: –12x3 + 106x2 – 241x + 4477Broccoli: 14x2 – 14x + 1545

In each polynomial, x = 0 corresponds to the first year of the 6 year period. What polynomial models the total number of pounds, in millions, of carrots and celery and broccoli consumed in the U.S. over the 6 year period?

Adding PolynomialsSimplify each.

1. (5r3 + 8) + (6r3 + 3)

2. (6x2 + 7) + (3x2 + 1)

3. (3z3 – 4z + 7z2) + (8z2 – 6z – 5)

4. (2k2 – k + 3) + (5k2 + 3k – 7)

PracticeSimplify:

Subtracting PolynomialsSimplify each.

1. (x3 – 3x2 + 5x) – (7x3 + 5x2 – 12)

2. (–4m3 – m + 9) – (–4m2 + m – 12)

3. (x2 – 2) – (3x + 5)

4. (14h4 + 3h3) – (9h4 +2h3)

5. (–9r3 + 2r – 1) – (–5r2 + r + 8)

6. (y3 – 4y2 – 2) – (6y3 + 4 – 6y2)

PracticeSimplify:

PracticeA local deli kept track of the sandwiches it sold for three months. The polynomials below model the number of sandwiches sold, where s represents day.

Ham & Cheese: Pastrami:

Write a polynomial that models the total number of these sandwiches that were sold.

PracticeA small town wants to compare the number of students enrolled in public and private schools. The polynomials below show the enrollment for each:

Public School:Private School:

Write a polynomials for how many more students are enrolled in public school than private school.

PracticeThe fence around a quadrilateral-shaped pasture is long. Three sides of the fence have the following lengths: . What is the length of the fourth side of the fence?

PracticeThe perimeter of a triangular park is . Two of the sides are: .

What is the sum of the two given sides?What operation should you use to find the remaining side

length?

Questions• Is it possible to write a trinomial with a degree of 0?

• Is the sum of two trinomials always a trinomial?

Exit TicketSimplify and Write in standard form:

Name the polynomial and the degree:

Multiplying and Factoring

Objective: To multiply a monomial by a polynomial. To factor a monomial from a polynomial.

Objective• I can multiply a monomial and a trinomial.

• I can find the greatest common factor.

• I can factor out a monomial.

• I can factor a polynomial model.

Vocabulary• You can use distributive property to multiply a monomial by a

polynomial.

Multiplying a Monomial and a TrinomialSimplify:

1. –x3(9x4 – 2x3 + 7)

2. 5n(3n3 – n2 + 8)

3. 4x(2x3 – 7x2 + x)

4. –8y3(7y2 – 4y – 1)

5. 3m2(10 + m – 4m2)

6. –w2(5w4 + 7w2 – 15)

PracticeSimplify each product.

VocabularyFactoring a polynomial reverses the multiplication process. When factoring a monomial from a polynomial, the first step is to fine the greatest common factor (GCF) of the polynomial’s terms.

Once you find the GCF of a polynomial’s terms, you can factor it out of the polynomial.

Finding the Greatest Common FactorFind the GCF of each.

1. 12x + 202. 8w2 – 16w3. 45b + 274. 5x3 + 25x2 + 45x5. 3x4 – 9x2 – 12x6. 4x3 + 12x – 28 7. a3 + 6a2 – 11a8. 14z4 – 42z3 + 21z2

PracticeFind the GCF of the terms of each polynomial.

Factoring Out a MonomialWhat is the factored form of each?

1. t2 + 8t2. 9x – 6 3. 4x5 – 24x3 + 8x4. 9x6 + 15x4 + 12x2

5. –6x4 – 18x3 – 12x2

6. 14n3 – 35n2 + 287. 5k3 + 20k2 – 158. g4 + 24g3 + 12g2 + 4g

PracticeFactor each polynomial.

Factoring a Polynomial Model• A helicopter landing pad, or helipad, is sometimes marked with a

circle inside a square so that it is visible from the air. What is the area of the shaded region of the helipad? Write your answer in factored form.

• Suppose the side length of the square is 6x and the radius of the circle is 3x. What is the factored form of the area of the shaded region?

x

2x

Practice• A circular mirror is surrounded by a square metal frame. The radius of

the mirror is 5x. The side length of the metal frame is 15x. What is the area of the metal frame? Write your answer in factored form.

• A circular table is painted yellow with a red square in the middle. The radius of the table is 6x. The side length of the red square is 3x. What is the area of the yellow part of the tabletop? Write your answer in factored form.

Multiplying BinomialsObjective: To multiply two binomials or a binomial by a trinomial.

Side Notes• There are three different methods that I will be showing you

on multiplying binomials.

• The three methods are: 1. FOIL, 2. Distributive, 3. Box.

• I want you to figure out which is easier for you and you work the problems that way. I do not care which method you use.

Objectives• I can use the distributive property.

• I can use a table.

• I can use the FOIL method.

• I can apply multiplication of binomials.

• I can multiply a trinomial and a binomial.

Using the Distributive PropertyWhat is the simplified form?

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

PracticeSimplify each product using the Distributive Property.

1. (x + 7)(x + 4)2. (y – 3)(y + 8)3. (m + 6)(m – 7)4. (c – 10)(c – 5)5. (2r – 3)(r + 1)6. (2x + 7)(3x – 4)

VocabularyWhen you use the Distributive Property to multiply binomials, notice that you multiply each term of the first binomial by each term of the second binomial.

A table can help you organize your work.

Using a TableWhat is the simplified form?

1. (5x – 3)(2x + 1)

2. (3x – 4)(x + 2)

3. (n – 6)(4n – 7)

4. (2p2 + 3)(2p – 5)

5. (x + 1)(x + 4)

6. (x + 2)(x + 4)

PracticeSimplify each product using a table.

1. (x + 5)(x – 4)2. (a – 1)(a – 11)3. (w – 2)(w + 6)4. (2h – 7)(h + 9)5. (x – 8)(3x + 1)6. (3p + 4)(2p + 5)

VocabularyFOIL stands forFirst – multiply the first term in both binomialsOuter – multiply the outer most terms in both binomialsInner – multiply the inner most terms in both binomialsLast – multiply the last term in both binomials

Using FOILWhat is the simplified form?

1. (x + 7)(x + 4)

2. (y – 3)(y + 8)

3. (m + 6)(m – 7)

4. (c – 10)(c – 5)

5. (2r – 3)(r + 1)

6. (2x + 7)(3x – 4)

PracticeSimplify each product using the FOIL method.

1. (a + 8)(a – 2)2. (x + 4)(4x – 5)3. (k – 6)(k + 8)4. (b – 3)(b – 9)5. (5m – 2)(m + 3)6. (9z + 4)(5z – 3)7. (3h + 2)(6h – 5)8. (4w+13)(w+2)9. (8c – 1)(6c – 7)

Applying Multiplication of BinomialsA cylinder has the dimensions shown in the diagram. What is the polynomial in standard form that describes the total surface area of the cylinder?

X + 1

X + 4

Applying Multiplication of BinomialsWhat is the total surface area of a cylinder with radius x + 2 and height x + 4? Write your answer as a polynomial in standard form.

Practice1. What is the total surface area of the cylinder? Write your answer as

a polynomial in standard form.

2. The radius of a cylindrical gift box is (2x + 3) inches. The height of the gift box is twice the radius. What is the surface area of the cylinder? Write your answer in standard form.

X + 5

X + 2

Multiplying a Trinomial and a BinomialWhat is the simplified form?

1. (3x2 + x – 5)(2x – 7)

2. (2x2 – 3x + 1)(x – 3)

3. (x + 5)(x2 – 3x + 1)

4. (k2 – 4k + 3)(k – 2)

5. (2a2 + 4a + 5)(5a – 4)

6. (2g + 7)(3g2 – 5g + 2)

PracticeWhat is the simplified form?

1. (5x3 + 4x2 – 7x)(x – 2)

2. (8b – 3)(7b2 + b – 9)

3. (b2 + 3)(b2 – 4b + 5)

4. (2x2 + 4x – 3)(3x + 1)

5. (3c5 – 4c + 8)(4c2 – 3)

PracticeSimplify each product. Write in standard form.

Multiplying Special Cases

Objective: To find the square of a binomial and to find the product of a sum and difference.

Objectives• I can square a binomial.

• I can apply squares of binomials.

• I can find the product of a sum and difference.

Vocabulary• There are special rules you can use to simplify the square of a binomials or the product of a

sum and difference.

• Squares of binomials have the form (a + b)2 or (a – b)2.

• The square of a binomial:• Words: The square of a binomial is the square of the first term plus twice the product of the two

terms plus the square of the last term.• Algebra:

• (a + b)2 = a2 + 2ab + b2 (a – b)2 = a2 – 2ab + b2

• Examples:• (x + 4)2 = x2 + 8x + 16 (x – 3)2 = x2 – 6x + 9

Squaring a BinomialWhat is the simpler form of each product?

1. (x + 8)2

2. (2m – 3)2

3. (n – 7)2

4. (2x + 9)2

5. (g – 2)2

6. (3x + 1)2

7. (8 + r)2

PracticeWhat is the simpler form of each product?

1. (3s + 9)2

2. (4x – 6)2

3. (a – 8)2

4. (w + 5)2

5. (5m – 2)2

6. (2n + 7)2

7. (h + 2)2

8. (k – 11)2

Applying Squares of BinomialsA square outdoor patio is surrounded by a brick walkway as shown. What is the area of the walkway?

3 feet

3 feet

X feet

X feet

What if the brick walkway is 4 feet wide. What is the area?

Practice• The figures below are squares. Find an expression for the are of each shaded

region. Write your answers in standard form.

• A square green rug has a blue square in the center. The side length of the blue square is x inches. The width of green band that surrounds the blue square is 6 inches. What is the are of the square band?

x

x + 4 𝑥+3

x – 1

Vocabulary• The product of the sum and difference of the same two terms also

produces a pattern. Notice that the sum of 0 – ab and ba is 0, leaving .

• The Product of a Sum or Difference:• Words: The product of the sum and difference of the same two terms is the

difference of their squares.• Algebra:

• (a + b)(a – b) = a2 – b2

• Examples:• (x + 2)(x – 2) = x2 – 4• (2x + 6)(2x – 6) = 4x2 - 36

Finding the Product of a Sum and DifferenceWhat is the simplified form?

1. (x3 + 8)(x3 – 8)2. (x + 9)(x – 9)3. (6 + m2)(6 – m2)4. (3c – 4)(3c + 4)5. (v + 6)(v – 6)6. (5r + 9)(5r – 9)7. (2g + 9h)(2g – 9h)8. (2p2 + 7q)(2p2 – 7q)

PracticeSimplify each product.

1. (v + 6)(v – 6)2. (b + 1)(b – 1)3. (z – 5)(z + 5)4. (x – 3)(x + 3)5. (10 + y)(10 – y)6. (t – 13)(t + 13)

PracticeSimplify each product.

Factoring x2 + bx + c

Objective: To factor trinomials of the form

Objectives• I can factor where b > 0, c > 0.

• I can factor where b < 0, c > 0.

• I can factor where c < 0.

• I can apply factoring trinomials.

• I can factor trinomials with two variables.

Vocabulary• You can write some trinomials of the form as the product of two binomials.

• The coefficient of the trinomials term is 1.

• The coefficient of the trinomial’s x-term is the sum of the numbers in the binomials.

• The trinomial’s constant term is the product of the same of the same two numbers.

• To factor a trinomial of the form x2 + bx + c as the product of binomials, you must find two numbers that have a sum of b and a product of c.

Factoring x2 + bx + c where b > 0, c > 0What is the factored form?

PracticeWhat is the factored form?

1. k2 + 5k + 62. v2 + 12v + 203. x2 + 12x + 354. y2 + 6y + 5

VocabularySome factorable trinomials have a negative coefficient of x and a positive constant term. In this case, you need to inspect the negative factors of c to find the factors of the trinomial.

Factoring x2 + bx + c where b < 0, c > 0What is the factored form?

1. x2 – 11x + 242. y2 – 6y + 8

PracticeWhat is the factored form?

1. x2 – 7x + 102. t2 – 10t + 243. g2 – 18g + 454. q2 – 8q + 12

Vocabulary• When you factor trinomials with a negative constant term, you need

to inspect pairs of positive and negative factors of c.

Factoring x2 + bx + c where c < 0What is the factored form?

1. x2 + 2x – 15

PracticeWhat is the factored form?

1. c2 – 4c – 21 2. q2 + 3q – 54 3. w2 – 7w – 8 4. n2 – 3n – 10

Applying Factoring Trinomials1. The area of a rectangle is given by the trinomial x2 – 2x – 35. What

are the possible dimensions of the rectangle? Use factoring.

2. A rectangle’s area is x2 – x – 72. What are the possible dimensions of the rectangle? Use factoring.

Practice1. The area of a rectangular desk is given by the trinomial d2 – 7d – 18.

What are the possible dimensions of the desk? Use factoring.

2. The area of a rectangular rug is given by the trinomial r2 – 3r – 4. What are the possible dimensions of the rug? Use factoring.

Vocabulary• You can factor some trinomials that have more than one variable.

• A trinomial with two variables may be factorable if the first term includes the square of one variable, the middle term includes both variables, and the last term includes the square of the other variable.

Factoring a Trinomial with Two VariablesWhat is the factored form?

1. x2 + 6xy – 55y2

2. m2 + 6mn – 27n2

PracticeWhat is the factored form?

1. k2 + 5kn – 84n2

2. p2 – 8pq – 33p2

3. w2 – 14wz + 40z2

4. m2 – 3mn – 28n2

Factoring ax2 + bx + c

Objective: To factor trinomials of the form ax2 + bx + c.

Objectives• I can factor when ac is positive.

• I can factor when ac is negative.

• I can apply trinomial factoring.

• I can factor out a monomial first.

Vocabulary• You can write some trinomials of the form ax2 + bx + c as the products

of two binomials.

• To factor a trinomial of the form ax2 + bx + c, you should look for factors of the product ac that have a sum of b.

Factoring When ac Is PositiveWhat is the factored form?

1. 5x2 + 11x + 22. 6x2 + 13x + 5

PracticeWhat is the factored form?

1. 3d2 + 23d + 142. 2x2 + 13x + 63. 4p2 + 7p + 34. 6r2 – 23r + 205. 4n2 – 8n + 36. 8g2 – 14g + 3

Factoring When ac is NegativeWhat is the factored form?

1. 3x2 + 4x – 152. 10x2 + 31x – 14

PracticeWhat is the factored form?

1. 4w2 + w – 3 2. 5x2 + 19x – 4 3. 3z2 + 23z – 36 4. 2k2 – 13k – 24 5. 4c2 – 5c – 6 6. 6t2 + 7t – 5 7. 4d2 – 4d – 35 8. 2x2 – x – 3

Applying Trinomial Factoring1. The area of a rectangle is 2x2 – 13x – 7. What are the possible dimensions of

the rectangle? Use factoring.

2. The area of a rectangle is 8x2 + 22x + 15. What are the possible dimensions of the rectangle? Use factoring.

3. The area of a rectangular kitchen tile is 8x2 + 30x + 7. What are the possible dimensions of the tile? Use factoring.

Practice• The area of a rectangle is 6x2 – 11x – 72. What are the possible

dimensions of the rectangle? Use factoring.

• The area of a rectangular knitted blanket is 15x2 – 14x – 8. What are the possible dimensions of the blanket? Use factoring.

Vocabulary• To factor a polynomial completely, first factor out the GCF (greatest

common factor) of the polynomial’s terms.

• Then factor the remaining polynomial until it is written as the product of polynomials that cannot be factored further.

Factoring Out a Monomial FirstWhat is the factored form?

1. 18x2 – 33x + 122. 8x2 – 36x – 20

PracticeWhat is the factored form?

1. 12p2 + 20p – 8 2. 20w2 – 45w + 103. 8v2 + 34v – 30 4. 12x2 – 46x – 8 5. 6s2 + 57s + 72 6. 9r2 + 3r – 30 7. 30x2 + 14x – 8 8. 66k2 + 57k + 12

PracticeFactor each expression completely.

Factoring Special Cases

Objective: To factor perfect – square trinomials and the differences of two squares.

Objectives• I can factor a perfect-square trinomial.

• I can factor to find a length.

• I can factor a difference of two squares.

• I can factor out a common factor.

Vocabulary• You can factor some trinomials by “reversing” the rules for multiplying special case

binomials.

• Any trinomial of the form is a perfect–square trinomial because it is the result of squaring a binomial.

• Factoring Perfect – Square Trinomials:• Algebra: For every real number a and b:

• Examples:

Vocabulary• Here is how to recognize a perfect – square trinomial:• The first and the last terms are perfect squares.

• The middle term is twice the product of one factor from the first term and one factor from the last term.

Factoring a Perfect – Square TrinomialWhat is the factored form?

PracticeFactor each expression.

Factoring to Find a LengthDigital images are composed of thousands of tiny pixels rendered as squares. Suppose the area of a pixel is . What is the length of one side of the pixel?

You are building a square patio. The area of the patio is . What is the length of one side of the patio?

PracticeThe given expression represents the area. Find the side length of the square.

100𝑟2−220𝑟+121 64𝑟 2−144𝑟+81 25𝑟 2+30𝑟+9

Vocabulary• Recall that .

• So you can factor a difference of two squares, , as (a + b)(a – b).

• Factoring a Difference of Two Squares:• Algebra: For all real numbers a and b:

• Examples:

Factoring a Difference of Two SquaresWhat is the factored form?

Factoring a Difference of Two SquaresWhat is the factored form?

PracticeFactor each expression

1.

2.

3.4.5.

VocabularyWhen you factor out the GCF of a polynomial, sometimes the expression that remains is a perfect – square trinomial or the difference of two squares. You can then factor this expression further using the rules that we have already learned.

Factoring out a Common FactorWhat is the factored form?

PracticeFactor each expression.

Factoring by Grouping

Objective: To factor higher – degree polynomials by grouping.

Objectives• I can factor a cubic polynomial.

• I can factor a polynomial completely.

• I can find the dimensions of a rectangular prism.

Vocabulary• Some polynomials of a degree greater than 2 can be factored.

• You have factored trinomials of the form by rewriting bx as a sum of two monomials.

• You then grouped the terms in pairs, factored the GCF from each pair, and looked for a common binomial factor.

• This process is called factoring by grouping.

• You can extend this technique to higher – degree polynomials.

Factoring a Cubic PolynomialWhat is the factored form?

PracticeFactor each expression.

VocabularyBefore factoring by grouping, you may need to factor out the GCF of all the terms.

Factoring a Polynomial CompletelyWhat is the factored form?

PracticeFactor completely.

Vocabulary• You can sometimes factor to find possible expressions for length,

width, and height of a rectangular prism.

Finding the Dimensions of a Rectangular Prism

A toy is made of several bars that can fold together to form a rectangular prism or unfold to form a “ladder”. What expressions can represent the dimensions of the toy when it is folded up? Use factoring.

A rectangular prism has a volume of . What expressions can represent the dimensions of the prism? Use factoring.

Practice• Find expressions for the possible dimensions of each rectangular

prism.

• A trunk in the shape of a rectangular prism has a volume of . What expressions can represent the dimensions of the trunk?

𝑉=3 𝑦3+14 𝑦2+8 𝑦 𝑉=4𝑐3+52𝑐2+160𝑐

Factoring Polynomials1. Factor out the greatest common factor (GCF).

2. 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.

3. If the polynomial has four or more terms, group terms and factor to find common binomial factors.

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