FMPCIOUnit 2: Polynomials

Lesson 1: Multiplying & Dividing Monomials

3x5y2This is a term of a polynomial. The 3 is called the coefficIent, the x and y are calledthe variables, and the 5 and 2 are the exponents. The exponents must be wholenumbers.

Types of polynomials

# of Terms Example Type of Polynomial

I

2

3

4

DEGREE OF POLYNOMIALS: In general, degree refers to the highest sum of

exponents of any one term of a polynomial

Example 1: State the degree of each polynomial

a)x3+x2—x+ 1 b)x3÷x2y2+y2÷xy+1 c)m4-m3+m2n3+n2-2

degree: degree: degree:

Example 2: Simplify

a) (3a) (2a) b) (12x) (3X2) (-2x)

1 2x5c) (2m2n3)(np2) d)

3x2

12m5 28m2n5e)

24m3f)

— 12m2n3

g) (3x2)(2x2)3 I) (—3x2y5)(—4xy3)28x4y4

HW: 2.1 Worksheet Attached

2.1 Multiplying & Dividing Monomials Worksheet

1! Simplify,.a) (7x2X5x4) N (3aX4a) ic) (—6m3X9m2 ci) (5n2X2n)e) (—8y5)(—7y4) 1) (—7b3X4b) s) — (PX—60) h) (12p5)(—3p3X—’2p)

2! Simplify,a) (6x)(3y) b) (Sp2X4q) c) (—5m3)(—ln) ci) (2cjbX7a)e) (—9rXSrs3) .1) (4c4)(—I 1cr) ) (_2s12X.3st) h) (2b2c3Xbc2)

3+ Simplify.a) (—4abX2b2) b) (6m2n3X—2m3n4) c) (7a2b2X3ab3)ci) (—8p2q5—3pq2) e) (2xy)(3x2y)(4x3y) f) (—3ab)(2ab2Xa2b)

4. Simplify.

a’ i2x b’36,,,6 i

54a8ci’ 521,

‘ 3x2‘ 4rn2

C,—6a2 ‘

e) —24cd3 28in2n5 . —32aNø li’—6c4

A!

-7rn2n3g,

‘ 3p2q2

5. Simplify

a) 36r0s4 b -45x8y 45p9q68r2s2 ‘ 15x4y2 ‘ —14p3q2

ci) (3a4b2X6ab3) e) (4m4n2X6mn9a3b4 —3rn6n6 . (h2y3)(6x3y)

6. 1,) (—3a2bX2a4)2 c) (5y2(S)(334)Z

ci) (_4pq3)(._5p3)2 e) (3m2n3X_21flfl2)3 1) (5x2.v)N3xZ9P

“.

_________

b) c)

ci)j42xy2)! e) o

FMPCIOUnit 2: Polynomials

Lesson 2: Adding & Subtracting Monomials

In General: When adding and subtracting polynomials you can only add or subtractlike terms. This means that you can only add and subtract terms thathave the same variables raised to the same powers respectively.

Examøle 1: Simplify

a) (2x — 7y) + (8x — 3y) b) (5x + 11) — (2x + 4)

c) (4x2 — 7xy + 3) + (x2 — 5yx + 9) — (x2 + 6xy — 11)

d)Add: 7x2—6x+24x2—3x— 4

Example 2: Simplify and state the degree of the monomial.

a) (4a2 — 3b2) + (2a2 — b2) — (3a2 + 4) =

b) (x2y2—y2) — (4x2y + 3x2y)

—(y2 — 2x2y2 — 2y2x) =

HW: 2.2WorksheetAttached

2.2 Adding and Subtracting Monomials Worksheet

1. Simplify.a) (6a÷9)÷(4a—5) 1,) (2x—7y)+(Sx—3y)

c) (3m2+4)(7+9m2) d) (+ll)—(2p+4)

e) (17x2—9x)—(6x2—5x) 1) (7k÷31)—(i2k—$1)

2. Simplify.a) (3m2—Sm+9)+(8m2+2m—7)

b) (7a3— 2a2 ÷ Sa) + (—3a3+ 6a2 — 2a)

c)(x2—Sx÷6y)—(4x2+i5x—lly)

d) (512_13t+17)+(9t3_712+31_26)_(l612+51_8)

e) (2÷8a2—a3)+(2a3—3a2—8)—(12 —7a3+5a2)

f) (4x2—lx+3)—(x2—Sx+9)—(8x2+6x—li)

4, Sirnplifi, and state the degree of the polynomiELa) (5x2—3x2y+2y2x—y)—(2xy2—y—5x2+33’X2)

b) (p3q2÷ 7q2p2— 3g,) + (2q — 6p2q2— q2p3)

c) (3rn2—3rn+7)—(2n2--2n+7)

d) (2ab—2ac—2bc)÷(2ca÷2cb—2ba+3)

e) (xy + 3yx + 2x2y — 3xy2)— (2xy ± 2yx2)

5. Mia) 3x2—6x±7 b)7z3—6z2÷z—37x2—2x÷9 2z±4z2_z±4

6. SWtract.a) 5x2y—Jxy+2xy2 1)6rn2n2—4mn÷3rnn3

3x2v ÷ 4xy — 2rn2n2t2rntt — 4mn3

7. Sb plify, and state the degree of the polynomial:,a) (3x2—2y2)÷(y2—2x2)—(4x2÷2)

b)(m2—n2)—(n2—m2)÷(2n2+m2)

c) (4x2y— 2yx) + (3yx2 — 6xy2)— (3x2y2÷ 2y2x2—

ci) (a2b2—b2)— (3b2a+ 2a2b) —(b2— b2a2 — b2a)

FMPIOUnit 2: Polynomials

Lesson 3: Multiplying Polynomials Part I

Example I: Simplify:

a) 2x(3xy—5)

b) -5x3y(4xy2+ 7x) + llx4y

c) 11(2x+ 3y)—7(3y— llx)

d) 2x(3x2 — 5xy + y2) — 5y2(x—

2y)

HW: 2.3 Worksheet Attached

2.3 Multiplying A Monomial and a Polynomial Worksheet

1. Smiify.a) 4(5x2 + 10) b) 72a — 5) c) -(3k — 2k)

4) 12(2b2—.3b+9) e) _9(...5ffi2+i?fl_3) 1) 3(8p25p+7)

2. Simplify.a) 3(x+4)+7 b) $(2a—3)÷lJa c) 5(y+2)—7y

d) 4(lm—5)—13 e) —6(3p+2p)÷5p2 f) 7(5x—3y)—$3x

a) 12 5x—4) b) 3a(—7a+2) c) 6p(2p—q)

d) —15n2(6 — 9n) e) 7rn3(3nm +6) f) —8x2(5x ÷ ly)

4. Simplil.a) 3(x +2) + 2(x —6) b) 2(x +9) — 3(x + 7)c) 3(2a÷Wb—2c)—6(a—2b+Sc) d) 3(2m—4n+3)—5(—2rn+5n—I)

5, Simplii’.a) 3x2(x + y) +2x3(3x + 5y) b) 3a3(2a — Sb) — 4a3(2a + 3b

c) 5p2(4p — q) —8p2(2p —7q) 4) 6a3(—3a ÷ 7b —4) — 8a3(2a — 3b +7)6. In a hockey league, each team plays afi other teams 4 iimes. Ifthere are n teams,

wiite a formula for the total number ofgames to be played.

7. Simp1if.a) —2ab2(ab—a2+b) b)3x2y(xy2+xy—y)

c) —Sm2n(3mn÷nm2—n2) 4) 5x(x—y)—231(x+y —1)+y2

e) 2b(b—bc)—2c(b —c)÷(76c—4c2)) 7x(x2—y2)—2xy—2y(x2÷y2)

9. Simplity, and state the degree ofthe polynomiaL

a) 3aTh —?a(a —4)+(5ba2—13a)

I,) 6rn(2mn — n) — 3(m2n — 19mn) + 6n(—Smn — 7m2)

c) 4xy(x + 2y) — $x(2xy — 3y2 — 3y(3x2+ 7xy

4) 5s(3E2—7s+2)+2s(5s2+s—6)+9(s2+6s—4)

e) 4x(5x2

1) Sxy—3y(2x —y)÷7x(Sy —x)+9(x3—2y2)

10. Simpliê,a) (1 la + 3X9a) b) (3x — SX—2x) c) $ 5x(16x + 12)4) (5a2+ 2a — 3X—4a) e) I 2n(3n —7) 1’) 5x(2x2— 6x + 3)

FMPIO

a) (x+3)(x—2)

b) (2a—4)(a+2)

c) (x+3)2

d) (2x—4)2

Unit 2: PolynomialsLesson 4: Multiplying Polynomials II

Example 1: Simplify:

e) (x—3)(4x2—7x+12)

f) (x —7) (x + 8) (3x —4)

Example 2: Express the area of the shaded region below as a polynomial.

3x+2y -*E- 2x—y

+y

3x-y

HW: 2.4 Worksheet Attached

2.4 Multiplying Polynomials Worksheet

1. Find the product.a) (4x÷ lX2x+3) b) (5in—2)6m+ 1) C) (2c—3)4c—5)

d) (3x — 7)(3x + 7) e) 4r + 5X4t — 8) f) 8x — 2X3x 7)

2. Square the binomial.a) (g+h) b)(x+y) c)(i+n) d)(b+c)

e) (p...q) f) (k—I) g) (s•..f) Ii) (u—v)

3. Find the product.a) (3A +y)2 b) (x+5y)2 c) (3a÷2)2 d) (Sx—9)2

e) 4rn—7)2 f) (—4y—1l)2 a) (5t+iO)2 Ii) (_5_6b)Z

4. Find the product.a) (2÷y)(3x+y) b) (3a+bX5a—b) c) (2x—y)(3x+4y)

d) (+3j)(+4i) e) (Sin—2nXlm—n) f) (6x—2yX3x—7y)

5. Find the product.a) (3a + 5b)2 b) (2m — 7n)2 c) (6s — 2i)3 4) (4p — 3q)2

e) (8x—3y)2 f) (—3s—5t)2 g) (—5f÷4g)2 ii) (5y2+7z2)2

6. Find the product.a) (x—y)(x+y) b) (2m÷5n)(2rn—5n) C) (9a—4bX9a+4b)

d) 6x — 2yX6x + 2y e) ç7p + 2q)7p — 2q) fl —12m + 9X—12m —9)

7. Simplify.a) (x_3)x+2)+(2X—5XX I) b) (2x+4(3x—2)+(5X —2X3x—-4)

c) (2m÷3)(3rn÷5)+(rn—7X6m—3)d) (c+2dXc+d)—(C—2d)(C—d)

e) (2a — 5b)(3a + b) — (6a — bX4a + 7b) f) (3x — 2y)(5x —y) — (2x + 5’X3x —

y)

8. Simplify.a) (2x_3)2+(x+2)(3X_S) b) (s—2t)2--(s—31)(4S—S1)

c) (2x — y)(2x + y) — (x —2y)2 d) (7in — 2nX3in + 5n) — (4rn — I in)2

e) (5x —2y)2 — (5x + 2y)2 f) (2x2 + 3Xx2 — 5) — (3x2 ÷ 2)2

9. Find the product.

a) (3x — 1X2x2÷3 ....4) b) (m—.3X4m2—7m+ 12)

c) (2a—5X3a2+8a—9)d)(3p+2)(5p2—6p+2)

10. Find the product.a) (3x+4)(x—5X2X+8) b) (4p—3X2p—7X5P—6)

c) (3x ÷22x— 5)4x —3) d) (2x —5)(3x +4)2

e) 5a — 32t2a — 7) 5m —

11 * Find the perimeter and area ofeach shaded region.a)

_________

1,)

x-2yr’)1_jS+y

k+y .. H1

12. Simplify,a) 3(m—5)(m+2)c) 5(2p—73p--4)e) 3a(2a — SbX3a + b)

13. Simplify.a) 2(3x+lXx+5)÷4(2x+3XJx+5)b) 5(2m — 4X3m + 2)— 2(4m — lX3m —4)

c) 3(2x + SyX7x — $y) + 5(x ÷ 3y)(2x ÷ 4y)

d) 2(xy— 1)(xy÷ t)—5(xy—2)(xy+ 3)

e) 3(2a — 3bXa + 2b) — 2(3a — 3)2

1) 4(xy—2)2+3(xy+5)2

14. Simplify.a) (2x—$X3x+6)+(3x—2X4x÷9)—(Sx—3)(2x—7)

b) 2(3s÷5)(2s—2)—5(3s2+7s—9)—(s+6)2c) (2x+3X3x2_Sx+4)÷(x_4X2xZ+3x_7)

d) (3m+4Xm—4n—i)—(5m—2)(3m—6n—8)

e) (4y — 5)(3y +2)2—(3y + 2X4y — 5)2

f) 4(3x—2X5x2÷x÷6)—(2x+6X3x—1)2

rxty

Fx,2y

I

F..

I —

c)

Zx

x-y1

r C-’

b) 2(x÷yX3x—5y)ci) 4x(x—ly)(2x—3y)

0 3x(2x +

FMPCIOUnit 2: Polynomials

Lesson 5: Factoring a GCF

2 (x + y) = this process is called

________________________

2x + = this process is called

______________________

Where the — outside is the

______________

Example 1: Use algebra tiles to factor 2x + 8.

Step I Model the polynomial 2x + 8.

Step 2 Arrange the tiles into a rectangle.The total area of the tiles representsthe product.

Example 2: Use Algebra Tiles to factor the following (sketch a diagram of solution).

a) 3x + 9

b) 4x+1O

c) x2 + 5x

d) 3x2 + 6x

GCF for a Polynomial

GCF of the coefficients and the lowest power of any variable in every term.

Example 2: Find the GCF of the following:

a) 12, 18 b) 3x2y3,6x3y c) I 5x3, I Ox, 35x

Example 3: Factor

a) 12x+18 +

b) 3x3+5x2—x=

c) 36m4+24m2—60=

d) 5x(a—b)+7(a—b)=

e) 2m(2x+ 1)—5(2x+ 1)=

f) 6a(b—a)+4b(b—a)—7(b—a)=

HW: Pg.155#4,7-12,14,16

FMPCIOUnit 2: Polynomials

Lesson 6: Factoring Trinomials Part I

Example 1: Factor the following using algebra tiles.

a) x2 + 5x + 6

b) x2 + Sx +4

c) x2 — 5x + 6

d)x2+x—6

e) x2 — 3x — 10

Example 2: Factor Completely.

a)a2+12a+2O=

b) m2—18m+32=

c)n2+3n—40

d)k2—8y—20

e) 2x2+16x+14

f) 3m3-18m2+27m

g)x2+l3xy+12y2

h) a4-’- 10a2+21

) (x+y)2+6(x+y)+8

j) (x-i-2y)2—4(x+2y)—12

HW: Pg. 166#4,6,11,14,17,19-21

FMPIOUnit 2: Polynomials

Lesson 7: Factoring Trinomials Part 2

Example 1: Factor the following.

a) 2x2 + 7x +6 1. Multiply the outside coefficientstogether.

2. Find two numbers that multiply to thisproduct and add to the middlecoefficient.

3. Set up brackets with the firstcoefficient in the front position ofboth sets and the two numbers thatyou found in the last position.

4. Divide any common factors out ofeach set of brackets.

5. Check by expanding

b) 8x2+2x—3

c) 2x2+9x+4

d) 8x2+lOx+3

e) 3x2—13x—1O

Example 2: Factor Completely.

a) 6a3+26a2—20a

b) 4x2y—l7xy—15y

c) 21x2+25xy—4y2

d) 4x2 — 2lxy — 1 8y2 =

HW: Pg. 177 #12,13, 15, 18,19

FMPIOUnit 2 Polynomials

Lesson 8: Factoring a Difference of Squares

Expand the following:

(x+ 3)(x—3)

(x + 2) (x —2) =

(3x—5)(3x+ 5)=

What’s the pattern???????????????????????????????????????????

To factor we will use this pattern and work backwards

Example 1: Factor the foNowing

a)x2—9 b)81m2—16

c) 1 6m2 — 1 21n2 d) 6x2 — 150

e) 1 2x3 — 27x f) 63x2y2—

g) (5m — 2)2 —25

h) x4 — 13x2 + 36

i) Y4 — 5y2 — 36

HW: Pg.194#6, 7, 10-13, 20, 21

2.1 Multiplying & Dividing Monomials Worksheet Key

_____

La)35x’ b)l2tFc)—54m’ d)10n3 e)56 O—28b4g$t’ h)72p’ 2.a)1$xy b)32pqc)35mn d)14a6e)—45A? fl-44cd 2)63213 bc’ 3s)-Sab’b)-12rn5n7c)2:ia’b d)24jPq e)24x6y’ f)—6a4b44.a)4x’ b)9rn c)—9a’ 4)46 e)4cP f)4n2 g)864b)10p245a)2r432 b)—3x’ c4p’q d)2a’be)—$m’ f)6x5t’ 6.a)54x” b)—12a’°b c$Ss’°s’d)—lOOp’q3e)-24mn’ f)675xyt4 7a)&’y3b)—6? c4a263 d)2x4 e)—16rn’n’ fla’3b”

2.2 Adding and Subtracting Monomials Worksheet Key

K

. tsci tcc.1 ‘Jey. 1.fl÷4bnÜE—1oYO1—o—34rnp÷l— 4x fl—5k + 11, 2.a11n$ — 3m + 2

b)4d+4d+3aC)—3Z3—i+ I7y— 189 — 1St — I e)Sd — 18 0ic2 — Sr ÷ 5

3a)P= 15.Sn — 30 000 1,1547 500; 3125 000; 5280 0001

4,a)1(le — 6x& third b)p’q’ — 3p + Zq; fourth- 3m — 2,9 + 2n;sccond d)3;zero

e)—3xf + 2xy third LaWh’ — Sx ÷ 16b)9z’—2t+1 4s)2fl—7Xy+52Y’b)4mW - 6,wz + 7aut 7.s)-3x - - 2; second

b)3m’; second c)-Sfl + 7zy - - 6xy2;fourth

d)2d9 - 2a’b - 2ab’ - 2b1; fourth 8.Answersmay vary.

2.3 Multiplying A Monomial and a Polynomial Worksheet Key

Eerdses 3-4, page 85i.a)2C1t+40 b)14a—35 c)lók-24k2d)24b2— 36b + 108 e)45m2— 63m +27fl24 — 15p €21 2a)3x + 19 b)24 — 5a eflO — 2y4)28m —33 e)—13 — 12p fl4x — 21y3.a)60x2— 48x b)6a — 2hz2 e)12p1— 6pqd)135n’—90n e)21m4n+42rn3 f)—5’—40x34s)Sx —6 b)-x —3 c)42b — 36c dN6rn — 5Th ÷145.t)9x’ + 13x2yb)—2a4—27a36c)4p3+ Slp’q:d>..34a’ + 66a36— 80a’ 6.2n(n —1)7.sfl2a3b2—

24261_ 2ab3 b)3x’y3+ 3x3y2—

c)Sm2n’— Sm’,? — 15m3n2d$x2— — 7+2ye)2b3-2b2c-29+Shef)7x3-lx/ - fly - Zx’y - 2 8a)66 b)Z’

9s)8a36—72 + iSa; third b)21 mit’ — 33m’n; thir:

c)lxy’- iSx2y; third d)43s’+21s2—38s;thirde)5x3 14x2y— 53vtbirdf)SSxy — 15,2 + 2x’; secondiO.a)99ê + 27a b)1Ox — :6x2 c)40x2+ 30xd)—20a’ — 8a’ + 12a c)3.6n’ — 8AirfllOx’—30x2+15x ils)29p— i8q+29b)—29k—41—75 c)3x’+Sxy—9x—29— i4yd)Sa”- lOOa e)45x’—8x2y—2x’f)30m2n—m’—94mn 12.a)a+5b b)lOa÷4bc)a’÷b’ d)a’—a’b—ah’ —6’ 134 =2icrr÷n)14s)911 cm2 b$& c)8xr2— IGn d)3&+ 14n

2.4 Multiplying Polynomials Worksheet Key

Key: I.a)8x2+14x+3 b)30m2—7rn —2

c)8c2—22c+ 15 d)9x2—49 e)16t2— 12t—40 4

f)24x2— 62x + 14 2.a)g2 + 2gh + h2 b)x2 ÷ 2xy +

c)m2+2mn÷n2d)b2+2bc÷c2e)p1—2pq+q2f)k2 — 2k! + j2 g)s2 — 2sf + 12 h)u2 — 2uv +

3a)9x2+ 6xy + y2 b)x2 + lOxy + 25y2

c)9a2 + 1 2a + 4 d)25x2 — 90x + 81e)16rn2—56m+49 f)169+88y+ 121

g)25r2 — 1001 + 100 h)25 ÷ 60b + 36b24..a)6x2+Sxy÷y2b)15a2+2ab—b2c)6x2+5xy-4y2d)x2÷7xy+ 12y2e)35m2— I9mn + 2n2 f)18x2 — 48xy + 14y2

5.a)9a2 + 3Oab + 25b2 b)4m2 — 28mn + 49n2

c)36s2 — 24sf + 42 d)16p2— 24pq + 9q2e)64x2— 48xy + 9y2 f)9s2 + 30s1 + 2512

g)25f2—4Ofg + 6g2 h)25y4+ 70y2z1+ 49z2

6.a)x —y b)4m —25n c)81a —16b

d)36x2 — 4y2 e)49p2 — 4q2 f)144m2— 81

7a)3x2 -8x — I b)21x2 — 18x c)12m2— 26m + 36

d)6cd e)—18a2-•5lab+2b2f)9x2—26xy+7y28a)7x2— lix— 1 b)—3s2÷l3st — 1Ir2

c)3x2 + 4xy -- 5y2 d)5m2÷ lllmn — 131,12 e)—4Oxy

f)—7x4 — 1 9x2— 19 9.a )6x4 7x2 — I 5x + 4

b)4m —19m +33m-r36c)6a +a —58a÷45

d)15p3— 8p2—

6p + 4 9)aJEx3+ 2x2 — 128x — 160

b)40p3—218p2÷309p—126 )24x3—62x2--7x÷3O •

d)18x3÷3x2— 88x —80 e)50a3—235a2+ 228a —63

f)125rn3—150m2+60m—8 11a)iOx+2y;

4x2 + 2xy — 4$ b)14x + 6y; lOx2 + l4xy + 2y2

c)16x+ 18;12x2+38x+20 12.,a)3m2—9m—30

b)6x2 — 4xy — 10y2 c)30p2— 145p + 140

d)8x3 — 68x2y+ 84xy2 e)18a3 — 39a2b — 15ab2

f)12x3+ 48x2y+ 48xy213a)30x2+108x+70 b)6rn2—2m--48

c)52x2+ IOlxy — 609 d)—3x2y2— 5xy + 28

e)—12a2÷l5ab—20b2f)7x2y2+l4xy+91

14..a)8x2+ 57x — 69 b}-4s2— 39s — 11

c)8x3 — 6x2 — 26x + 40d)—12m2÷ISmn ÷47m —28n—20e)— 1 2$ + 91 v2 — 39y — 70 f)42x3 — 70x2 + 98x — 54

15a)m2+ tim + 26 b)3m2+ 16m + 28

c)rn3 + 5m2 + 6m — 5 16a)4a2+ 4ab + 4b2b)—5a2+ l2ab — 3b2 c)8a2 ÷ 6ab ÷ 2b2

ID:A

Polynomials ReviewAnswer Section

MULTIPLE CHOICE SHORT ANSWER

1. B 30. (s—32)(s---l)2. A 31. 7(2z—5)(z—l)3. B 32. 27z3+102z2+6Z+1204. B 33. (7s+8t)(7S—8t)5. A 8(z—5)(z—9)6. C

35. =4(7x—2y)(7x+2Y)7. C

8.A9. D 36. a) The outer rectangle area =(9y)(6y) = 54y2

10. C11.

The inner rectangle has area (5y)(3y) = 15y2

13. B b) The total area of the shaded region is the difference in the areas of the rectangles:14. C 54y2—15y2=39y215.

a)6m2—llm+3 a) 4y2—25 c) 6x2+l3xy+6y2 d) 2a4—a2—10 e) 8x2—16x+4

17 B 37. f) 2p2 —6p+4 g) —22m h) —6m2 —5mn+9m+4n+2n2 i) lOx2 —llx+3 j) 9x2 —4918. A k) 32x2+16x+2 1) 2p2—6p+4 m) 6x+20y

19. D a) (x+4)(x+3) b) (m—5)(m+2) c) (2k÷l)2 d) (2p—3)2 e) (6x—5y2)(6x+5y2)20. B f) (x—2)(x+2)(x2+4) g) 5(2x—3)2 h) (—x—1)(3x+1) i) (m—3n)(3m+4n) j) (x+2)(5x—1)21. B 38. k) (x—S)(x—4) 1) (2x—3)(3x+4) m) (x+3)2 n) (m—9)(m+2) o) (6k+1)222.

p) 2(p2 —6p+8) q) (3x—2y)(3x+2y)(9x2+2y2) r) (x-13)(x+13) s) (x—2)(4x—7) t) —4(y—l)(2y

24. Du) not factorable (prime) v) (x + 3)(x —4— Sy) w) (x

—y — 7)(x

— y + 2)

25.A26. B

27. B

28. B29. B

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Name: Class: Date: ID:A

U2Polynomials Review

Multiple ChoiceIdentjfr the choice that best completes the statement or answers the question.

1. Simplify the expression

y2 +8y—6—9y2—24y—26,then factor.

a. —8(y2—2y—4)

b. —8(y2+2y+4)

c. —4(2y2+4y+8)

d. —4(2y2+4y+1)

2. Expand and simplify: (p +3)(p —7)

a. p2—4p—21

b. p2—lOp—21

c. p2+lOp—21

d. p2+4p—21

3. Expand and simplify: (6p + 3)(5p —6)

a. 30p2+21p—18

b. 30p2—21p—18

c. 30p2+51p—18

d. 30p2—51p—18

4. Expand and simplify:(3c + 2)(2c —7) + 3(—2c + 1)(7c —5)

a. —36c2+8c—29

b. —36c2+34c—29

c. —36c2—8c—19

d. —36c2—8c—29

5. Expand and simplify: (2x2 + 5x — 6)(5x2 — 2x + 3)

a. lOx4 +21x3—34x2 +27x—18

b. 10x4+21x3—34x2—3x+18

c. lOx4 +21x3—24x2 +27x+ 18

d. lOx4 —29x3—34x2 +27x— 18

6. Expand and simplify: (8h + 3)(7h2 — 4h + 1)

a. 56h3 —53h2—20h+3

b. 56h3÷11h2—12h+3

c. 56h3—11h2—4h+3

d. 56h3—32h2+8h+3

7. Expand and simplify: (5m — 3n)2

a. 25m2—9n2

b. 25m2—lSmn+9n2

c. 25m2 —3Omn+9n2

d. 25m2+9n2

8. Each shape is a rectangle. Write a polynomial, insimplified form, to represent the area of the shadedregion.

a. 5x2+31x+66

b. 5x2+37x+30

c. 5x2+31x+30

d. 5x2+37x+66

9. Which polynomial, written in simplified form,represents the area of this rectangle?

8x — 4y

a. 8x2—36xy—20y2

b. 8x2+22xy—20y2

c. 16x2+72xy—40y2

d. 8x2+36xy—20y2

x + 5y

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Name: ID:A

10. Factor the binomial 44a + 99a2.a. a(44+99a)

b. 11(4a+9a2)c. lla(4+9a)d. 22a(2+9a)

11. Factor the trinomial —33b2 + 99b + 77.

a. —11(3b2—9b+7)

b. —33(b2—3b—7)

c. —11(3b2—9b--7)

d. 33(—b2+27b÷7)

12. Factor the trinomial —24c3d — 40c2d2 — 32cd3.

a. —8cd(3c2— 5cd—4d2)

b. 8cd(3c2+5cd+4d2)

c. 8cd(—3c2+5cd+4d2)

d. —8cd(3c2+5cd+4d2)

13. Identify the greatest common factor of the terms in34 42 23

the trmomial 6s t + 1 2s t — 15s t22a. 6st

b. 3s2t2

c. 3s2t3

d. 3s3t2

14. Factor:t2+9t—36a. (t—2)(t+ 18)b. (t+2)Q—18)c. (t+12)(t—3)d. (t—12)Q÷3)

215. Factor: —84+8z+za. (42+z)(—2+z)b. (—6+z)(14+z)c. (—42+z)(2+z)d. (6+z)(—14+z)

16. Factor: —4d2 — 28d + 240a. —4(d + 3)(d — 20)b. —4(d+5)(d—12)c. —4(d—3)(d+20)d. —4(d—5)(d+12)

17. Complete. (k—D)(k—5)=k2—Dk+135

a. (k—27)(k—22)=k2—5k+135

b. (k—27)(k—5)=k2—32k+135

c. (k—27)(k—32)=k2—5k÷135

d. (k-.27)(k—5)=k2—22k+135

18. Factor: 7n2 — 14n —105a. 7(n+3)(n—5)b. 7(n+15)(n—1)c. 7(n—15)(n+1)d. 7(n—3)(n+5)

19. Which multiplication sentence does this set ofalgebra tiles represent?

_H_____H____

a. (2x—2)(2x+2)

b. (2+2)(2÷2)

c.d. (2x-f-2)(2x+2)

20. Factor: 25x2 ÷ 58x + 16a. (25x+4)(x+4)b. (25x+8)(x+2)c. (5x+4)(5x+4)d. (5x+8)(5x+2)

21. Factor: 48y2—

ll6y+6Oa. (l6y—l2)(3y—5)b. 4(4y—3)(3y—S)c.d. 4(4.y+3)(3y+5)

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Name: ID:A

22. Factor: 96w2 + 324w —42a. 6(8w+1)(2w—7)b. 6(8w+7)(2w—1)c. 6(8w—7)(2w+1)d. 6(8w—1)(2w+7)

23. Factor: 121a2+llOa+25a. (lla+5)(lla—5)b. (121a+5)(a+5)

c. (ha—S)2

d. (lla+5)2

24. Factor: 16p2—81q2

a. (4p—9q)2

b. (4p+9q)2c. (16p—9q)(p—9q)

d. (4p+9q)(4p—9q)

25. Factor: 3z4 — 768z2

a. 3z2(z+ 16)(z— 16)

b. 3z2(z+ 16)2

c. z2(z+48)(z—16)

d. 3z2(z—16)2

26. Factor Completely: 162 — 2w4

a. (9—w2)(18—w2)

b. 2(9+w2)(3+w)(3—w)

Written

c. 2(9—w2)2

d. 2(9+w2)2

30. Factor: S2 —33s+32

31. Factor: 14z2—49z+35

32. Expand and simplify: (9z2 — 2z + 10)(3z + 12)

33. Factor: 49s2 — 64t2

27. Determine the area of the shaded region in factoredform.

a. 4(x+12)b. (3x+2)(x+12)c. (3x+12)(x+2)

d. (3x—2)(x—12)

28. For which values of k can 6x2 +k+1 befactored?

a. , 7 only

b. ±5, ±7 only

c. -5, -7 only

d. all integers between -7 and 5, inclusive

29. Which of the following is a factor of the

polynomial 3x2 + 2x —8?

a. 3x+2b. 3x-4c. 3x+4d. x-2

34. Factor. Check by expanding.

8z2 — 112z+ 360

35. Factor.

196x2 — 16y2

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Name: ID:A

37. Simplify the following:

(2m—3)(3m—1)

(2y—5)(2y+5)

(2x+3y)(3x+2y)

(a2 +2)(2a2 5)

4(2x—1)2

2(p —1) (p —2)

5(—2m--3n)—3(4m—5n)

—3m (2m + n —3)— 2n (m — n —2)

(5x—3)(2x—1)

(3x+7)(3x—7)

) 2(4x+1)

1) 2(p—1)(p—2)

m) —3(2x—4y)+4(3x+2y)

38. Factor Completely

a) x2+7x+12

6) m2-3m-1O

c) 4k2+4k+1

d) 4p2—12p+9

e) 36x2—25y4

f) x”—16

g) 20x2+60x+45

6) x2_(2x+1)2

9 3m2 —5mn—12n2

j) 5x2+9x—2k) x(x—5)—4(x—5)

6x2—x—12

m) x2+6x+9

n) m2—7x—18

o) 36k2+12k+1

p) 2p2—12p+16

q) 81x4—16y4

r) x2—169

s) 4x2—15x+14

t) y2_(3y_2)2

u) 9m2—l8mn—8n2v) x(x+3)—4(x+3)+5y(x+3)

w) (x—y)2—5(x—y)—14

36. The diagram below shows one rectangle insideanother.

a) Determine the area of each rectangle.b) Determine the area of the shaded region.

a)

6)

c)

e)

I)g)

6)

9

4

ID:A

Polynomials ReviewAnswer Section

MULTIPLE CHOICE

1. B2.A3. B4. B5.A6. C7. C8.A9.D

10. C11. C12.D13. B14. C15. B16.D17. B18.A19.D20. B21. B22.D23.D24.D25.A26. B27. B28. B29. B

SHORT ANSWER

30. (s—32)(s—1)

31, 7(2z—5)(z—1)

32. 27z3 + 102z2 +6z+ 120

33. (7s+8t)(7s—8t)

34. 8(z—5)(z—9)

35. =4(7x—2y)(7x+2y)

ID:A

36. a) The outer rectangle area = (9y)(6y) = 54y2

The inner rectangle has area = (5y)(3y) = 15y2

b) The total area of the shaded region is the difference in the areas of the rectangles:

54y2—15y2=39y2

a) 6m2 —llm+3 a) 4y2 —25 c) 6x2 +l3xy+6y2 d) 2a4 —a2 —10 e) 8x2 —16x+4

37. f) 2p2—6p+4 g) —22m h) —6m2—5mn+9m+4n+2n2i) lOx2 —llx+3 j) 9x2—49

k) 32x2+16x+2 1) 2p2—6p+4 m) 6x+20y

a) (x+4)(x+3) b) (m—5)(m+2) c) (2k+l)2 d) (2p—3)2 e) (6x—5y2)(6x+5y2)

f) (x—2)(x+2)(x2+4) g) 5(2x—3)2 h) (—x—1)(3x+l) i) (m—3n)(3m+4n) j) (x+2)(5x—l)

38. k) (x—5)(x—4) 1) (2x—3)(3x+4) m) (x+3)2 n) (m—9)(m+2) o) (6k+l)2

p) 2(p2 —6p+8) q) (3x—2y)(3x+2y)(9x2+2y2) r) (x-13)(x+13) s) (x—2)(4x—7) t) —4(y—l)(2y—l)

u) not factorable (prime) v) (x+3)(x—4—5y) w) (x—y—7)(x--y+2)

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