~ Chapter 9 ~ Polynomials and Factoring Algebra I Lesson 9-1 Adding & Subtracting Polynomials Lesson 9-2 Mulitplying and Factoring Lesson 9-3 Multiplying

~ Chapter 9 ~Polynomials and Factoring

Algebra I

Lesson 9-1 Adding & Subtracting Polynomials

Lesson 9-2 Mulitplying and Factoring

Lesson 9-3 Multiplying Binomials

Lesson 9-4 Multiplying Special Cases

Lesson 9-5 Factoring Trinomials of the Type x2 + bx + c

Lesson 9-6 Factoring Trinomials of the Type ax2 + bx + c

Lesson 9-7 Factoring Special Cases

Lesson 9-8 Factoring by Grouping

Chapter Review

Algebra I

Adding & Subtracting

Polynomials Cumulative Review Chap 1-8

Lesson 9-1

Adding & Subtracting PolynomialsNotesLesson 9-1

Monomial – an expression that is a number, variable, or a product of a number and one or more variables. (Ex. 8, b, -4mn2, t/3…)

(m/n is not a monomial because there is a variable in the denominator)

Degree of a Monomial

¾ y Degree: 1 ¾ y = ¾ y1… the exponent is 1.

3x4y2 Degree: 6 The exponents are 4 and 2. Their sum is 6.

-8 Degree: 0 The degree of a nonzero constant is 0.

5x0 Degree = ?

Polynomial – a monomial or the sum or difference of two or more monomials.

Standard form of a Polynomial…

Simply means that the degrees of the polynomial terms decrease from left to right.

5x4 + 3x2 – 6x + 3 Degree of each?

The degree of a polynomial is the same as the degree of the monomial with the greatest exponent. What is the degree of the polynomial above?


3x2 + 2x + 1 12 9x4 + 11x 5x5

The number of terms in a polynomial can be used to name the polynomial.

Classifying Polynomials

(1)Write the polynomial in standard form.

(2) Name the polynomial based on its degree

(3) Name the polynomial based on the number of terms

6x2 + 7 – 9x4 3y – 4 – y3 8 + 7v – 11v

Adding Polynomials

There are two methods for adding (& subtracting) polynomials…

Method 1 – Add vertically by lining up the like terms and adding the coefficients.

Method 2 – Add horizontally by grouping like terms and then adding the coefficients.

(12m2 + 4) + (8m2 + 5) =


(9w3 + 8w2) + (7w3 + 4) =

Subtracting Polynomials

There are two methods for subtracting polynomials…

Method 1 – Subtract vertically by lining up the like terms and adding the opposite of each term in the polynomial being subtracted.

Method 2 – Subtract horizontally by writing the opposite of each term in the polynomial being subtracted and then grouping like terms.

(12m2 + 4) - (8m2 + 5) =

(30d3 – 29d2 – 3d) – (2d3 + d2)

Adding & Subtracting Polynomials

HomeworkLesson 9-1

Homework – Practice 9-1

Multiplying & Factoring Practice 9-1Lesson 9-2

Multiplying & Factoring

Practice 9-1Lesson 9-2



Mulitplying & FactoringNotesLesson 9-2

Distributing a monomial

3x(2x - 3) = 3x(2x) – 3x(3) =

-2s(5s - 8) = -2s(5s) – (-2s) (8) =

Multiplying a Monomial and a Trinomial

4b(5b2 + b + 6) = 4b(5b2) + 4b(b) + 4b(6) =

-7h(3h2 – 8h – 1) =

2x(x2 – 6x + 5) =

Factoring a Monomial from a Polynomial

Find the GCF for 4x3 + 12x2 – 8x

4x3 = 2*2*x*x*x

12x2 = 2*2*3*x*x

8x = 2*2*2*x What do they all have in common? 2*2*x = 4x

Multiplying & FactoringNotesLesson 9-2

Find the GCF of the terms of 5v5 + 10v3

Find the GCF of the terms of 4b3 – 2b2 – 6b

Factoring out a Monomial

Step 1: Find the GCF

Step 2: Factor out the GCF…

Factor 8x2 – 12x =

Factor 5d3 + 10d =

Factor 6m3 – 12m2 – 24m =

Factor 6p6 + 24p5 + 18p3 =


HomeworkLesson 9-2

Homework ~ Practice 9-2 even

Multiplying Binomials Practice 9-2Lesson 9-3

Multiplying BinomialsNotesLesson 9-3

Using the Distributive Property

Simplify (6h – 7)(2h + 3) = 6h(2h + 3) – 7(2h + 3) =

(5m + 2)(8m – 1) = 5m(8m – 1) + 2(8m - 1) =

(9a – 8)(7a + 4) = 9a(7a + 4) – 8(7a + 4) =

Multiplying using FOIL

F = First O = Outer I = Inner L = Last

(6h – 7)(2h + 3) = 6h(2h) + 6h(3) + (-7)(2h) + (-7)(3)

12h2 + 18h + (-14h) + (-21) = 12h2 + 4h -21

(3x + 4)(2x + 5) =

(3x – 4)(2x – 5) =

Applying Multiplication of Polynomials

Determine the area of each rectangle and subtract the area of center

(x + 8)(x + 6) = 3x(x + 3) =

Multiplying BinomialsNotesLesson 9-3

Multiplying a Trinomial and a Binomial

(2x – 3)(4x2 + x -6) = 2x(4x2) + 2x(x) + 2x(-6) -3(4x2) -3(x) -3(-6)

8x3 + 2x2 + (-12x) - 12x2 -3x + 18

Combine like terms = 8x3 – 10x2 – 15x + 18

You can also multiply using the vertical multiplication method…

Try this one…

(6n – 8)(2n2 + n + 7) =

Multiplying Binomials

HomeworkLesson 9-3

Homework – Practice 9-3 even

Multiplying Special Cases




Multiplying Special CasesNotesLesson 9-4

Finding the Square of a Binomial

(x + 8)2 = (x + 8)(x + 8) =

So… (a + b)2 =

Rule: The Square of a Binomial

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

(a – b)2 = a2 – 2ab + b2

Find (t + 6)2

(5y + 1)2

(7m – 2p)2

Find the Area of the shaded region…

(x + 4)2 – (x – 1)2

Mental Math – Squares

312 = (30 + 1)2 = 302 + 2(30*1) + 12 = 900 + 60 + 1 = 961

Multiplying Special CasesNotesLesson 9-4

292 =

982 =

Difference of Squares

(a + b)(a – b) = a2 – ab + ab – b2

= a2 – b2

Find each product.

(d + 11)(d – 11) = d2 – 112 = d2 – 121

(c2 + 8)(c2 – 8) =

(9v3 + w4)(9v3 – w4) =

Mental Math

18 * 22 = (20 + 2)(20 – 2) = 202 – 22 = 400 – 4 = 396

59 * 61 =

87 * 93 =


Homework

Lesson 9-4

Homework – Practice 9-4

odd

Factoring Trinomials of the Type x2 + bx + c





NotesLesson 9-5

Factoring Trinomials

x2 + bx + c

To factor this type of trinomial… you must find two numbers that have a sum of b and a product of c.

Factor x2 + 7x + 12

Make a table…

Column 1 lists factors of c

12…

Column 2 lists the sum of those factors… b

Row 3 – factors 3 & 4 with a sum of 7 fits so…

x2 + 7x + 12 = (x + 3)(x + 4)

Factor g2 + 7g + 10

Factor a2 + 13a + 30


NotesLesson 9-5

Factoring x2 – bx + c

Since the middle term is negative, you must find the negative factors of c, whose sum is –b.

Factor d2 – 17d + 42 > Make a table…

Row 3 – factors -3 & -14

with sum of -17

So… d2 – 17d + 42 = (d – 3)(d – 14)

Factor k2 – 10k + 25

Factor q2 – 15q + 36

Factoring Trinomials with a negative c (- c)

Factor m2 + 6m - 27

Make a table

Row 4 – factors 9 & -3 with sum of 6


NotesLesson 9-5

So… m2 + 6m – 27 = (m + 9)(m – 3)

Factor p2 – 3p – 40

Factor m2 + 8m – 20

Factor y2 – y - 56


HomeworkLesson 9-5

Homework ~ Practice 9-5 #1-30

Factoring Trinomials of the Type ax2 + bx + c


Factoring Trinomials of the Type ax2 + bx + cPractice 9-5

Lesson 9-6

Factoring Trinomials of the Type ax2 + bx + c


Factoring Trinomials of the Type ax2 + bx + cNotes

Lesson 9-6

Factoring Trinomials when c is positive

6n2 + 23n + 7… Multiply a & c

So… 6n2 + 2n + 21n + 7

Factor using GCF

2n(3n + 1) + 7(3n + 1)

(2n + 7)(3n + 1) = 6n2 + 23n + 7

Try another one… 2y2 + 9y + 7

So… 2y2 + 2y +7y + 7

Factor… 2y(y + 1) + 7(y + 1)

(2y + 7)(y + 1)

What if b is negative? 6n2 – 23n + 7

6n2 - 2n – 21n + 7

Factors of a*c

Sum (=b)

6 and 7 13

3 and 14 17

2 and 21 23 √


Lesson 9-6

2n(3n - 1) – 7(3n – 1)

(2n – 7)(3n – 1)

Your turn… 2y2 – 5y + 2

Factoring Trinomials when c is negative…

7x2 – 26x – 8

7x2 -28x + 2x – 8

7x(x – 4) + 2(x – 4)

(7x + 2)(x – 4)

Factor 5d2 – 14d – 3

5d2 -15d + 1d – 3

5d(d – 3) + 1(d – 3)

(5d + 1)(d - 3)

Factors of a*c

Sum (=b)

7 and -8 -1

4 and -14 -10

2 and -28 -26 √


Lesson 9-6

Factoring Out a Monomial First

20x2 + 80x + 35

Factor out the GCF first…

5(4x2 + 16x + 7)… then factor 4x2 + 16x + 7

4x2 + 2x + 14x + 7

2x(2x + 1) + 7(2x + 1)

(2x + 7)(2x + 1) Remember to include the GCF in the final answer

5(2x + 7)(2x + 1)

Factor 18k2 – 12k - 6

6(3k2 – 2k – 1)

3k2 - 3k + 1k – 1

3k(k – 1) + 1(k – 1) = 6(3k + 1)(k - 1)

Factoring Trinomials of the Type ax2 + bx + cHomework

Lesson 9-6

Homework: Practice 9-6 first column

Factoring Special Cases





NotesLesson 9-7

Perfect Square Trinomials

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

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

So… x2 + 12x + 36 = (x + 6)2

And… x2 – 14x + 49 = (x – 7)2

What about… 4x2 + 12x + 9

Factoring a Perfect-Square Trinomial with a = 1 (ax2 + bx + c)

x2 + 8x + 16 =

n2 – 16n + 64 =

Factoring a Perfect-Square Trinomial with a ≠ 1

9g2 – 12g + 4

4t2 + 36t + 81


NotesLesson 9-7

Factoring the Difference of Squares

a2 – b2 = (a + b)(a – b)

Or… x2 – 16 =

What about 25x2 – 81 =

Try x2 – 36

Factor 4w2 – 49

Look for common factors…

10c2 – 40 =

28k2 – 7 =

3c4 – 75 =


HomeworkLesson 9-7

Homework: Practice 9-7 odd

#1-39

Factoring by Grouping







NotesLesson 9-8

Factoring a Four-Term Polynomial

4n3 + 8n2 – 5n – 10

Factor the GCF out of each group of 2 terms.

? (4n3 + 8n2) - ? (5n + 10)

Factor 5t4 + 20t3 + 6t + 24

Before you factor, you may need to factor out the GCF.

12p4 + 10p3 -36p2 – 30p

Try… 45m4 – 9m3 + 30m2 – 6m (factor completely)

Finding the dimensions of a rectangular prism

The volume (lwh) of a rectangular prism is 80x3 + 224x2 + 60x. Factor to find the possible expressions for the length, width, and height of the prism.


NotesLesson 9-8

Your turn…

Find expressions for possible dimensions of the rectangular prism…

V = 6g3 + 20g2 + 16g

V = 3m3 + 10m2 + 3m


HomeworkLesson 9-8

Classwork – Practice 9-8 even

# 1-28



Chap 9 Quiz Review Lesson 7 & 8

Practice 9-8

~ Chapter 9 ~Chapter Review

Algebra I Algebra I

~ Chapter 9 ~Chapter Review

Algebra I Algebra I

Documents

~ Chapter 9 ~ Polynomials and Factoring Algebra I Lesson 9-1 Adding & Subtracting Polynomials Lesson 9-2 Mulitplying and Factoring Lesson 9-3 Multiplying