Section 3.3 Common Factors of a PolynomialEx. Factor: 8x4 – 10x3y + 6x3y2 (II) Factoring Trinomials of the Form x2 + bx + c Ex. Factor: y2 – 9y – 36 What if the order is reversed?

Math 1201 Unit III Factoring 1

Section 3.3 – Common Factors of a Polynomial

Goals:

Review of GCF

How to Attain a GCF between Monomials with Variables

How to Remove a GCF for a Polynomial with Variables



Practice Questions:

P.155 – P.156 #4, #8, #15b, #16


Practice Sheet – Removing a GCF from Polynomial

Identify the G.C.F. and express in factored form by removing the G.C.F.

1. 30x3y – 18x2y2 2. –14x4 + 21x3

3. 12a3b2 – 8a2b3 + 16ab2 4. 24p2q2 – 16pq3 – 12p2q

5. –27x4 + 18x3 + 9x2 6. 16p2 – 32pq + 24q2

7. – 50c4d2 + 75cd2 8. 36x3y – 8xy2 – 16x2y

9. 12x3 + 20x + 8x2 10. 81c4d2 – 36cd + 45c2d3


3.5 – Polynomials of the form x2 + bx + c

Goals:

Multiplying binomials

Factoring x2 + bx +c trinomials

ANSWERS:

1. 6x2y(5x – 3y) 2. –7x3(2x – 3) 3. 4ab2(3a2 – 2ab + 4)

4. 4pq(6pq – 4q2 – 3p) 5. –9x2(3x2 – 2x – 1)

6. 8(2p2 – 4pq + 3q2) 7. –25cd2(2c3 – 3)

8. 4xy(9x2 – 2y – 4x) 9. 4x(3x2 + 5 + 2x)

10. 9cd(9c3d – 4 + 5cd2)


(I) Multiplying Binomials

To multiple binomials we can use the following methods:

Method 1: Use a rectangle diagram.

 Example: (x – 3)(x + 1)

Method 2: Use the distributive property.

  Example: (x – 3)(x + 5)

Expand and simplify.

(a) (x – 4)(x + 2) (b) (8 – b)(3 – b)


Example: Expanding binomials by algebra tiles

Which algebra tile model best represents the expansion of (𝑥 + 4)(𝑥 + 3)?

A.

B.

C.

D.

(II) Factoring x2 + bx + c Trinomials

To factor a trinomial such as: x2 – 12x + 20

Expand: (x – 2)(x – 10)

We will expand the binomials first

(x – 2)(x – 10)

and look at the four terms that form the

trinomial above.

Note:

The two middle terms ____ and ____ ADD to give ____

Also, these two numbers ___ and ___ MULTIPLY to give ___


To factor a trinomial such as:

x2 – 12x + 20 into the product of two binomials, two conditions must

be satisfied.

So the trinomial below factors as:

x2 – 12x + 20 =

We can check by expanding ( )( ) to see if it generates

the trinomial above.

Example: Factor each trinomial

(a) x2 – 2x – 8 (b) p2 – 12p + 35

The two numbers must MULTIPLY to give ___

The two numbers must ADD to give ___

What number combination satisfies both conditions?

ANSWER: ______ AND ______


(II) Factoring x2 + bx + c Trinomials – continued

Factoring Trinomials written in ascending order.

Example: Factor

(a) –24 – 2q + q2 (b) 40 + 3r – r2

Factoring a trinomial with a common factor.

Example: Factor

(a) –3t2 + 3t + 36

(b) 4x3 – 44x2 + 120x (c) –6m2 + 18m + 60

Procedure:

Identify and remove the G.C.F. first

Factor the remaining trinomial

P.167

#14b, d, f, h #15 #19a, f #21


3.6  Factoring Trinomials of the Form ax2 + bx + c

(I) Expanding Binomials Algebraically

Example: Determine the area for the rectangle

Expanding by F.O.I.L. (First Outside Inside Last)

(II) Expanding Binomials Visually by Algebra Tiles

Example: Expand (2x + 1)(x + 3)

Goals:

Expanding binomials (ax + b)(cx + d)

Factoring ax2 + bx +c trinomials by Trial and Error

Dimensions

Dimensions Area =


(III) Factoring Trinomials of the Form ax2 + bx + c

If we are given area and required to produce the dimensions of the

rectangle

then the previous process is reversed as we produce the factors of a

trinomial ax2 + bx + c.

Summary:

Expanding binomials (ax + b)(cx + d) by algebra

tiles or F.O.I.L. produces a trinomial of the form

__________________


(A) Factoring Trinomials by Algebra Tiles

Example: Factor Using Algebra Tiles

Area = 2x2 + 7x + 3

Factored Form of 2x2 + 7x + 3 =

Procedure:

(i) Using algebra tiles construct a

rectangle that represents the area

2x2 + 7x + 3

(ii) From the rectangle of part (i)

determine the binomial

dimensions by reading the edge

length and edge width of the

rectangle.

Dimensions

Dimensions


(B) Factoring Using Trial and Error Method

Example: Factor 2x2 + 7x + 3

2x2 + 7x + 3

Step I

State all pairs of

factors for the first

and last terms

Step II

Determine what factors from the

first term that must multiply off

factors of the last term so that

each result will add up to the

middle term.

Step III

Use the correct arrangement for

multiplication in Step II to write

the two binomials.

Step IV

Check the order of the

arrangement to see if expansion

would produce the given

trinomial.


Example: Factor each of the following:

(a) 6x2 + x – 2 (b) 4x2 – 4x – 15 (c) 5x2 – 2x + 2

Practice Questions:

P.177 - 178 #5b, d #9a, d #10b, e #13c, d, h, #15 d, e, f, g, h


Factoring Review for Exit Card

(I) Removing a GCF

Ex. Factor: 8x4 – 10x3y + 6x3y2

(II) Factoring Trinomials of the Form x2 + bx + c

Ex. Factor: y2 – 9y – 36

What if the order is reversed? Ex. Factor: 24 – 5x – x2

(III) Factoring Trinomials of the Form x2 + bx + c with GCF removal First

Ex. Factor –2p4 – 6p3 + 56p2

The first thing to consider when factoring is to identify if

there is a ______ that can be removed.

Two things to consider for these trinomials:

What numbers multiply to give ‘c’ ?

The same numbers that multiply to

give ‘c’ must add to produce ‘b’


(IV) Factoring Trinomials of the Form ax2 + bx + c

Ex. Factor: 20x2 – 7x – 6

Completely factor each of the following:

1. –20q3 + 12q2 – 8q4 2. y2 – 12y + 27

3. 8x2 – 14x – 15 4. –3p3 – 9p2 + 30p

5. 18 + 3x – x2 6. 15y2 – 16y + 4

7. 4x2 + 4x – 24 8. x2 – 7x – 30


3.6 continued –  Factoring Trinomials of the Form ax2 + bx + c

Example: Factor completely.

1. 12x2 – 2x – 4

Goal:

Factoring Trinomials ax2 + bx + c where G.C.F is removed first

Step I

Identify the ______ first

Step II

Factor out the ______

_____ =

Step III

Factor remaining ______

ANSWERS:

1. –4q2(5q – 3 + 2q2) 2. (y – 9)(y – 3) 3. (4x + 3)(2x – 5)

4. –3p(p + 5)(p – 2) 5. (6 – x)(3 + x) or –(x – 6)(x + 3)

6. (5y – 2)(3y – 2) 7. 4(x + 3)(x – 2) 8. (x + 3)(x – 10)


Example: Factor completely.

2. 15x4 – 39x3 – 18x2 3. –12p3 – 22p2 + 20p

  3.7 Multiplying Polynomials

Recall: Distributive Property a(b + c) =

 when multiplying polynomials, every term from the first polynomial

must be multiplied off every term in the second polynomial

Example: Expand and simplify:

1. (3g + 2)(2g – 1) 2. (2r + 5t)2

Practice Questions:

P.178 #18 #19a, c, e, g, i #20a, c, e #21

Goal:

Multiplying polynomials by the distributive property


3. (2h + 5)(h2 + 3h – 4) 4. (–3f2 + 3f – 2)(4f2– f – 6)

5. (2c – 3)(c + 5) + 3(c – 3)(–3c + 1)

6. (3x + y – 1)(2x – 4) – (3x + 2y)2

7. Determine the expression (in expanded form) that represents the area of the

shaded region.

x + 6 2x

x + 5

3x + 2


3.8 Factoring Special Polynomials

(I) Factoring Perfect Square Trinomials

Factor:

1. 4x2 + 12x + 9

2. 4 – 20x + 25x2

What relationship do you notice about the factors and the trinomial?

Practice Questions:

P.186 #4 #5a, d, e, f, #7a (i), (ii)  #8a, d, #9c

Goals:

Identifying and factoring perfect square trinomials

Factoring the Difference of Two Squares

Factoring Trinomials in Two Variables


Example: Identify and factor each perfect square trinomial?

1. 25x2 + 60x + 36 2. 16x2 – 24x + 9

3. 2x2 + 6x – 8

(II) Factoring The Difference of Two Squares

Perfect Square Trinomials:

Produce the ___________ binomial factors.

First and last terms are ____________ square terms.

Pattern for factoring:

a2 + 2ab + b2 = ( )( ) OR ( )2

a2 – 2ab + b2 = ( )( ) OR ( )2

Difference of Two Squares:

Consist of ______ terms that are both _________ square terms.

The algebraic operation between the terms is _______________.

Pattern for factoring:

a2 – b2 = ( )( )


Example: Factor the following:

1. x2 – 25 2. h2 – 64 3. x2 + 36

Example: Factor each binomial:

1. 25 – 36x2 2. 5x4 – 80y4

(III) Factoring Trinomials in Two Variables

Example: Factor

1. 2a2 – 7ab + 3b2 2. 10c2 – cd – 2d2


Unit 3 Factoring Review

1. Expand (3x + 4y)(5x – 2y)

2. Factor completely:

(a) 18 + 7x – x2 (b) 2y3 – 8y2 – 42y

(c) 36 – 49x2 (d) 48y2 – 75

Practice Questions:

P.194  #5  #6  #8 #13

Key Ideas:

Expanding binomials by F.O.I.L

For ANY FACTORING PROBLEM always look to remove a ______ first

Factoring the difference of two squares by pattern

a2 – b2 = ( )( )

Factoring perfect square trinomials:

a2 + 2ab + b2 = ( )( ) or ( )2

a2 – 2ab + b2 = ( )( ) or ( )2


(e) 15x2 – 14xy – 8y2 (f) 18x3 + 3x2 – 36x

(g) 4x2 – 20xy + 25y2

3. Expand and express in simplest form:

(a) (3x – 4y)2 (b) (2p + 3)(4p2 – 5p + 2)

4. Determine the area for:

x + 3

2x + 5

7x – 2

3x + 4


Unit Review Factoring

Sections (3.3 – 3.8)

1. What is the greatest common factor in the trinomial 3 2 2 4 312 20 16a b a b a b ? 1.____

(A) 4 (B) 4ab (C) 24a b (D) 4 34a b

2. In the diagram, algebra tiles are used to help model the product. Which

polynomial represents the area modeled? (Shaded tiles are positive)

2.____

(A) 2 3x x

(B) 2 6x x

(C) 22 3x x

(D) 22 6x x

3. Expand: 5 6 4 3d d 3. ____

(A) 220 39 18d d (B) 220 9 18d d

(C) 220 9 18d d (D) 220 39 18d d

4. A polynomial is represented by the tiles shown below. What are the factors of the polynomial?

(Shaded tiles are positive, unshaded tiles are negative) 4.____

(A) 2 3x x

(B) 2 3x x

(C) 2 3x x

(D) 2 3x x


5. Completely factor the binomial 215 18y y . 5.____

(A) 23 5 6y y (B) 3 5 6y y

(C) 15 18y y (D) 3 15 18y y

6. The area of a rectangle is 2 2 24 x x . What are the dimensions of the rectangle? 6.____

(A) 4 6x by x (B) (x – 4) by (x + 6)

(C) 4 6x by x (D) (x + 4) by (x + 6)

7. Factor: 2 16x . 7.____

(A) 2

4x (B) 2

4x

(C) 4 4x x (D) 2 2 4x x x

8. What are the correct factors of 36 – 5x – x2 ? 8.

(A) ( 9 – x )( 4 + x ) (B) ( 9 – x )( 4 – x )

(C) ( 9 + x )( 4 – x ) (D) ( x – 9 )( x + 4 )

9. What are the factors of 210 3 1x x 9.____

(A) 5 1 2 1x x (B) 5 1 2 1x x

(C) 5 1 2 1x x (D) 5 1 2 1x x

10. Which represents a perfect square trinomial? 10.___

(A) 4x2 + 10x + 25 (B) 9x2 + 24x + 16

(C) 36 – 9x + x2 (D) x2 + xy + y2

11. Expand and simplify:

(a) (4x – 5y)2 (b) (x – 2)(x2 + 2x + 4)

(c) 3x(2x – 3)2 – 5(x + 2)(x – 2)

12. Determine the area of the shaded region in expanded form.

x + 2

x + 4

5x + 6

4x + 3


13. You plan to put siding on the front of your garage pictured below. Find an expression (in

simplest form) to represent the area of the surface to be covered with siding. (Note: There

will be NO siding on the two doors.)

14. Factor completely each of the following expressions.

(a) 12 + x – x2 (b) 2p3–12p2 + 16p

(c) 8x2 –10x – 3 (d) 63x2 – 33x – 6

(e) 49y2 – 81 (f) 25p2 – 30p + 9

(g) 10a2 + 11ab – 6b2 (h) 200x2 – 18

(i) 16u2 + 20u – 6 (j) w2 –3w – 28

(k) –18x3 – 15x2 + 18x (l) 3y2 –9y – 30

Review Problems from textbook:

P.198 – P.200

#11(f), #12(e), #18(g), #19(b), (d), (f), #24(c), #25(b), (d), (f)

#32, #33(a), (c), (e), #34


ANSWERS:

1. C 2. D 3. C 4. B 5. B 6. C 7. C 8. C 9. B 10. B

11(a) 16x2 – 40xy + 25y2 (b) x3 – 8 (c) 12x3 – 41x2 + 27x + 20

12. 19x2 + 33x + 10 13. x2 + 8x + 11

14.(a) (3 + x)(4 – x) (b) 2p(p – 4)(p – 2)

or –(x + 3)(x – 4)

(c) (4x + 1)(2x – 3) (d) 3(7x + 1)(3x – 2)

(e) (7y – 9)(7y + 9) (f) (5p – 3)2

(g) (2a + 3b)(5a – 2b) (h) 2(10x – 3)(10x + 3)

(i) 2(2u + 3)(4u – 1) (j) (w + 4)(w – 7)

(k) –3x(3x – 2)(2x + 3) (l) 3(y – 5)(y + 2)

Documents

Section 3.3 Common Factors of a PolynomialEx. Factor: 8x4 – 10x3y + 6x3y2 (II) Factoring Trinomials of the Form x2 + bx + c Ex. Factor: y2 – 9y – 36 What if the order is reversed?